linear algebra. session 9 - texas a&m universityroquesol/math_304_fall_2019_session_9.pdf ·...
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![Page 1: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Linear Algebra. Session 9
Dr. Marco A Roque Sol
10 / 25 / 2018
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 2: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/2.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 3: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/3.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section
we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 4: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/4.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again
a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 5: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/5.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n
linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 6: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/6.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations
with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 7: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/7.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 8: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/8.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 9: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/9.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where
the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 10: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/10.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix
A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 11: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/11.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A
is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 12: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/12.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued.
If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 13: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/13.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek
solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 14: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/14.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutions
of the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 15: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/15.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form
x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 16: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/16.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt ,
then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 17: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/17.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then
it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 18: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/18.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that
λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 19: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/19.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be
an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 20: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/20.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand
v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 21: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/21.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v
a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 22: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/22.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector
of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 23: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/23.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the
coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 24: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/24.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 25: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/25.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case,
λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 26: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/26.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ
is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 27: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/27.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex,
we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 28: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/28.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have
complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 29: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/29.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues and
eigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 30: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/30.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors
always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 31: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/31.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear
in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 32: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/32.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate.
Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 33: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/33.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus,
if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 34: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/34.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 35: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/35.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν;
v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 36: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/36.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 37: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/37.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 38: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/38.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate
eigenvalues and eigenvectors of thematrix A, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 39: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/39.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors
of thematrix A, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 40: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/40.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors of thematrix A,
then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 41: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/41.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 42: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/42.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 43: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/43.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued
solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 44: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/44.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions,
but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 45: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/45.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but
taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 46: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/46.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 47: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/47.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t =
eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 48: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/48.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 49: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/49.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition,
then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 50: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/50.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 51: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/51.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)
X2(t) =1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 52: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/52.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 53: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/53.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two (real) solutions !!!
X1(t) = eµt (acos(νt)− bsin(νt))
X2(t) = eµt (acos(νt) + bsin(νt))
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 54: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/54.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two
(real) solutions !!!
X1(t) = eµt (acos(νt)− bsin(νt))
X2(t) = eµt (acos(νt) + bsin(νt))
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 55: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/55.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two (real) solutions !!!
X1(t) = eµt (acos(νt)− bsin(νt))
X2(t) = eµt (acos(νt) + bsin(νt))
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 56: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/56.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two (real) solutions !!!
X1(t) = eµt (acos(νt)− bsin(νt))
X2(t) = eµt (acos(νt) + bsin(νt))
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 57: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/57.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
are two (real) solutions !!!
X1(t) = eµt (acos(νt)− bsin(νt))
X2(t) = eµt (acos(νt) + bsin(νt))
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 58: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/58.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.6
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 59: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/59.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.6
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 60: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/60.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.6
Solve
the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 61: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/61.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.6
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 62: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/62.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.6
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 63: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/63.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.6
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 64: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/64.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.6
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 65: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/65.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.6
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find
the eigenvalues of the matrix A
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 66: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/66.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.6
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues
of the matrix A
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 67: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/67.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.6
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix
A
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 68: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/68.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.6
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 69: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/69.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
|A− λI| =
∣∣∣∣∣∣3− λ 1 1
0 2− λ 10 −1 2− λ
∣∣∣∣∣∣ = 0
(3− λ)
∣∣∣∣2− λ 1−1 2− λ
∣∣∣∣ =
(3− λ)(λ2 − 4λ+ 5) = 0 =⇒
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 70: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/70.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
|A− λI| =
∣∣∣∣∣∣3− λ 1 1
0 2− λ 10 −1 2− λ
∣∣∣∣∣∣ = 0
(3− λ)
∣∣∣∣2− λ 1−1 2− λ
∣∣∣∣ =
(3− λ)(λ2 − 4λ+ 5) = 0 =⇒
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 71: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/71.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
|A− λI| =
∣∣∣∣∣∣3− λ 1 1
0 2− λ 10 −1 2− λ
∣∣∣∣∣∣ = 0
(3− λ)
∣∣∣∣2− λ 1−1 2− λ
∣∣∣∣ =
(3− λ)(λ2 − 4λ+ 5) = 0 =⇒
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 72: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/72.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
|A− λI| =
∣∣∣∣∣∣3− λ 1 1
0 2− λ 10 −1 2− λ
∣∣∣∣∣∣ = 0
(3− λ)
∣∣∣∣2− λ 1−1 2− λ
∣∣∣∣ =
(3− λ)(λ2 − 4λ+ 5) = 0 =⇒
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 73: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/73.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
|A− λI| =
∣∣∣∣∣∣3− λ 1 1
0 2− λ 10 −1 2− λ
∣∣∣∣∣∣ = 0
(3− λ)
∣∣∣∣2− λ 1−1 2− λ
∣∣∣∣ =
(3− λ)(λ2 − 4λ+ 5) = 0 =⇒
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 74: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/74.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 75: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/75.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = 2,
λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 76: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/76.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2=
2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 77: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/77.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 78: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/78.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 79: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/79.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 80: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/80.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 81: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/81.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 82: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/82.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 83: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/83.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
100
If λ2 = 2 + i , then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 84: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/84.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and
a corresponding eigenvector is
v(1) =
100
If λ2 = 2 + i , then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 85: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/85.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding
eigenvector is
v(1) =
100
If λ2 = 2 + i , then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 86: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/86.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
100
If λ2 = 2 + i , then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 87: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/87.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
100
If λ2 = 2 + i , then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 88: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/88.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
100
If
λ2 = 2 + i , then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 89: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/89.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
100
If λ2 = 2 + i ,
then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 90: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/90.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
100
If λ2 = 2 + i , then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 91: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/91.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
100
If λ2 = 2 + i , then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 92: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/92.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
100
If λ2 = 2 + i , then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 93: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/93.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
3− (2 + i) 1 10 2− (2 + i) 10 −1 2− (2 + i)
v1v2v3
=
1− i 1 10 −i 10 −1 −i
v1v2v3
=
1− i 1 10 −i 10 0 0
v1v2v3
=
1− i 0 1− i0 −i 10 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 94: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/94.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
3− (2 + i) 1 10 2− (2 + i) 10 −1 2− (2 + i)
v1v2v3
=
1− i 1 10 −i 10 −1 −i
v1v2v3
=
1− i 1 10 −i 10 0 0
v1v2v3
=
1− i 0 1− i0 −i 10 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 95: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/95.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
3− (2 + i) 1 10 2− (2 + i) 10 −1 2− (2 + i)
v1v2v3
=
1− i 1 10 −i 10 −1 −i
v1v2v3
=
1− i 1 10 −i 10 0 0
v1v2v3
=
1− i 0 1− i0 −i 10 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 96: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/96.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
3− (2 + i) 1 10 2− (2 + i) 10 −1 2− (2 + i)
v1v2v3
=
1− i 1 10 −i 10 −1 −i
v1v2v3
=
1− i 1 10 −i 10 0 0
v1v2v3
=
1− i 0 1− i0 −i 10 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 97: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/97.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
3− (2 + i) 1 10 2− (2 + i) 10 −1 2− (2 + i)
v1v2v3
=
1− i 1 10 −i 10 −1 −i
v1v2v3
=
1− i 1 10 −i 10 0 0
v1v2v3
=
1− i 0 1− i0 −i 10 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 98: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/98.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 99: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/99.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and
a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 100: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/100.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding
eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 101: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/101.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 102: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/102.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+
i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 103: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/103.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 104: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/104.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding
solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 105: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/105.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions
of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 106: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/106.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation
are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 107: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/107.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 108: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/108.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ;
x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 109: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/109.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 110: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/110.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 111: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/111.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 112: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/112.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 113: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/113.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t 6= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 114: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/114.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian
of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t 6= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 115: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/115.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these solutions
is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t 6= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 116: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/116.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t 6= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 117: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/117.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t 6= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 118: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/118.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t 6= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 119: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/119.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t 6= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 120: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/120.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)=
e7t 6= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 121: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/121.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t
6= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 122: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/122.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t 6= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 123: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/123.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 124: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/124.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence,
the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 125: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/125.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions
x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 126: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/126.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions x(1), x(2) and
x(3) form a fundamental set,and the general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 127: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/127.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions x(1), x(2) and x(3)
form a fundamental set,and the general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 128: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/128.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and
the general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 129: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/129.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution
of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 130: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/130.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system
is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 131: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/131.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 132: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/132.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is
X =
c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 133: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/133.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 134: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/134.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X =
c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 135: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/135.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t +
c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 136: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/136.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 137: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/137.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 138: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/138.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 139: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/139.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 140: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/140.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 141: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/141.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 142: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/142.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 143: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/143.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 144: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/144.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is
the direction field associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 145: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/145.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field
associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 146: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/146.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field associated with
the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 147: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/147.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 148: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/148.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 149: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/149.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 150: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/150.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 151: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/151.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 152: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/152.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 153: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/153.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.7
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 154: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/154.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.7
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 155: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/155.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.7
Solve
the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 156: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/156.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.7
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 157: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/157.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.7
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 158: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/158.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.7
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 159: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/159.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.7
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 160: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/160.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.7
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find
the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 161: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/161.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.7
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues
of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 162: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/162.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.7
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix
A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 163: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/163.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.7
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 164: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/164.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.7
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 165: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/165.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.7
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 166: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/166.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.7
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 =
(λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 167: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/167.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Example 9.7
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 168: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/168.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 169: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/169.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = −1
2+ i ,
λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 170: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/170.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 171: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/171.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If
λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 172: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/172.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If λ1 = −12 + i ,
then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 173: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/173.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 174: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/174.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 175: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/175.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 176: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/176.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=
(−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 177: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/177.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 178: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/178.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
(1i
)If λ2 = −1
2 − i , then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 179: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/179.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and
a corresponding eigenvector is
v(1) =
(1i
)If λ2 = −1
2 − i , then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 180: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/180.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding
eigenvector is
v(1) =
(1i
)If λ2 = −1
2 − i , then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 181: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/181.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
(1i
)If λ2 = −1
2 − i , then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 182: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/182.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
(1i
)
If λ2 = −12 − i , then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 183: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/183.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
(1i
)If
λ2 = −12 − i , then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 184: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/184.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
(1i
)If λ2 = −1
2 − i ,
then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 185: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/185.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
(1i
)If λ2 = −1
2 − i , then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 186: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/186.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
(1i
)If λ2 = −1
2 − i , then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 187: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/187.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
(1i
)If λ2 = −1
2 − i , then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 188: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/188.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 189: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/189.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 190: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/190.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and
a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 191: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/191.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding
eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 192: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/192.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 193: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/193.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)
The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 194: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/194.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions
of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 195: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/195.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation
are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 196: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/196.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 197: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/197.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ;
x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 198: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/198.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 199: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/199.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 200: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/200.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 201: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/201.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain
a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 202: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/202.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of
real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 203: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/203.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions,
we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 204: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/204.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose
the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 205: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/205.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand
imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 206: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/206.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts
of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 207: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/207.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or
x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 208: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/208.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2).
In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 209: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/209.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 210: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/210.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 211: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/211.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 212: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/212.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+
i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 213: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/213.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)
Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 214: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/214.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence,
a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 215: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/215.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of
real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 216: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/216.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued
solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 217: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/217.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions
are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 218: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/218.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 219: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/219.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)
v(t) = e−t/2(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 220: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/220.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence, a set of real-valued solutions are
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 221: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/221.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these two real-valued solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence, the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 222: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/222.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian
of these two real-valued solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence, the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 223: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/223.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these two
real-valued solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence, the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 224: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/224.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these two real-valued solutions
is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence, the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 225: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/225.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these two real-valued solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence, the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 226: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/226.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these two real-valued solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence, the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 227: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/227.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these two real-valued solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence, the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 228: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/228.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these two real-valued solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t
6= 0
Hence, the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 229: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/229.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these two real-valued solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence, the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 230: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/230.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these two real-valued solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence,
the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 231: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/231.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these two real-valued solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence, the solutions
x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 232: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/232.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these two real-valued solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence, the solutions x(1),
x(2) form a fundamental set,
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 233: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/233.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these two real-valued solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence, the solutions x(1), x(2)
form a fundamental set,
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 234: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/234.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these two real-valued solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence, the solutions x(1), x(2) form a
fundamental set,
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 235: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/235.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
The Wronskian of these two real-valued solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence, the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 236: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/236.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 237: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/237.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and
the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 238: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/238.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and the general solution
of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 239: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/239.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and the general solution of the system
is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 240: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/240.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 241: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/241.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and the general solution of the system is
X =
c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 242: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/242.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) =
c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 243: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/243.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+
c2e−t/2
(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 244: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/244.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)
Here is the direction field associated with the system(x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 245: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/245.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is
the direction field associated with the system(x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 246: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/246.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field
associated with the system(x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 247: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/247.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with
the system(x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 248: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/248.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system
(x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 249: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/249.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 250: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/250.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 251: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/251.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 252: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/252.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 253: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/253.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 254: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/254.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 255: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/255.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Complex Eigenvalues
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 256: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/256.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 257: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/257.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude
our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 258: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/258.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration
of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 259: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/259.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous system
with constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 260: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/260.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 261: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/261.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 262: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/262.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion
of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 263: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/263.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case
in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 264: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/264.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix
A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 265: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/265.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A
has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 266: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/266.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues.
suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 267: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/267.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that
λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 268: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/268.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ
is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 269: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/269.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root
of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 270: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/270.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the
characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 271: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/271.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation
∣∣∣A− λI∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 272: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/272.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 273: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/273.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then,
λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 274: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/274.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ
is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 275: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/275.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue
of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 276: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/276.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity
2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 277: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/277.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2
of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 278: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/278.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA.
In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 279: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/279.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event,
there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 280: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/280.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are
two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 281: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/281.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities:
The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 282: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/282.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A
isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 283: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/283.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and
there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 284: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/284.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and
there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 285: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/285.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still
a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 286: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/286.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions
ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 287: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/287.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 288: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/288.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,
weλt}
.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 289: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/289.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then, λ is an eigenvalue of algebraic multiplicity 2 of the matrixA. In this event, there are two possibilities: The matrx A isnon-defective and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 290: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/290.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 291: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/291.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However,
if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 292: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/292.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A
is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 293: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/293.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective,
there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 294: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/294.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is
just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 295: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/295.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution
ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 296: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/296.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form
veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 297: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/297.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt
associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 298: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/298.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with
this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 299: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/299.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue.
Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 300: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/300.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore,
toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 301: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/301.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution,
it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 302: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/302.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary
to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 303: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/303.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find
other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 304: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/304.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solution
of a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 305: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/305.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 306: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/306.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that
a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 307: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/307.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation
occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 308: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/308.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for
the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 309: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/309.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0
when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 310: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/310.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation
had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 311: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/311.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r .
In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 312: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/312.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case
we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 313: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/313.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found
one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 314: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/314.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution
y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 315: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/315.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,
but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 316: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/316.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second
independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 317: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/317.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution
had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 318: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/318.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form
y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 319: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/319.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 320: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/320.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way,
it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 321: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/321.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural
to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 322: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/322.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find
a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 323: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/323.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a second
independent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 324: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/324.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution
of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 325: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/325.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 326: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/326.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation had a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 327: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/327.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 328: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/328.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but,
doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 329: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/329.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and
substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 330: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/330.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system
we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 331: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/331.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find that
w = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 332: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/332.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus,
we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 333: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/333.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 334: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/334.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 335: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/335.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and
substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 336: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/336.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting
this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 337: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/337.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x
in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 338: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/338.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system
we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 339: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/339.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find
the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 340: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/340.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 341: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/341.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 342: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/342.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 343: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/343.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation
is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 344: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/344.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved
with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 345: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/345.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and
only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 346: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/346.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one
is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 347: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/347.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining
to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 348: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/348.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 349: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/349.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Example 9.8
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 350: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/350.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Example 9.8
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 351: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/351.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Example 9.8
Find
the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 352: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/352.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Example 9.8
Find the solution
of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 353: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/353.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Example 9.8
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 354: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/354.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Example 9.8
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 355: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/355.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Example 9.8
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 356: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/356.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Example 9.8
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 357: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/357.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Example 9.8
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find
the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 358: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/358.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Example 9.8
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues
of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 359: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/359.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Example 9.8
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix
A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 360: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/360.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Example 9.8
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 361: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/361.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Example 9.8
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 362: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/362.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Example 9.8
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 363: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/363.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Example 9.8
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 364: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/364.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 365: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/365.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
λ1 = 2,
λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 366: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/366.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 367: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/367.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If
λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 368: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/368.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2,
then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 369: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/369.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 370: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/370.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 371: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/371.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 372: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/372.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 373: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/373.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)
and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 374: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/374.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and
a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 375: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/375.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding
eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 376: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/376.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 377: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/377.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 378: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/378.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 379: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/379.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and
the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 380: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/380.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution
is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 381: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/381.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 382: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/382.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 383: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/383.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now,
for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 384: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/384.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second
solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 385: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/385.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution
we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 386: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/386.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 387: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/387.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t +
ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 388: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/388.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 389: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/389.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where
u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 390: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/390.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u
satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 391: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/391.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 392: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/392.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u =
(A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 393: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/393.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u =
v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 394: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/394.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 395: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/395.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 396: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/396.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 397: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/397.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 398: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/398.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 399: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/399.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so
if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 400: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/400.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k ,
where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 401: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/401.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k
is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 402: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/402.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary,
then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 403: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/403.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1.
If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 404: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/404.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 405: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/405.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 406: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/406.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 407: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/407.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 408: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/408.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)
then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 409: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/409.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then
by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 410: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/410.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting
for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 411: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/411.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and
u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 412: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/412.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u,
we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 413: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/413.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 414: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/414.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 415: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/415.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 416: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/416.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t +
ke2t(
1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 417: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/417.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + ke2t
(1−1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 418: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/418.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 419: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/419.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above
is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 420: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/420.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely
a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 421: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/421.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple
of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 422: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/422.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solution
x (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 423: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/423.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and
may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 424: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/424.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored,
but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 425: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/425.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but
the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 426: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/426.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first
two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 427: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/427.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms
constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 428: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/428.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute
anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 429: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/429.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 430: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/430.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 431: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/431.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 432: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/432.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 433: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/433.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation
shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 434: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/434.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) =
− e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 435: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/435.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and
therefore{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 436: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/436.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 437: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/437.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}
form a fundamental set of solutions ofthe system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 438: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/438.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set
of solutions ofthe system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 439: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/439.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions
ofthe system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 440: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/440.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system.
The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 441: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/441.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution
is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 442: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/442.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 443: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/443.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x =
c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 444: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/444.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) +
c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 445: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/445.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 446: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/446.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 447: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/447.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 448: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/448.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 449: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/449.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again
the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 450: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/450.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 451: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/451.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 452: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/452.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and
suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 453: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/453.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that
r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 454: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/454.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ
is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 455: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/455.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue
of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 456: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/456.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A,
but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 457: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/457.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one
corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 458: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/458.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v.
Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 459: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/459.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then
one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 460: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/460.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 461: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/461.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλt
where v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 462: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/462.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere
v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 463: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/463.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v
satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 464: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/464.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 465: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/465.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 466: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/466.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and
a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 467: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/467.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution
is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 468: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/468.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 469: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/469.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 470: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/470.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where
u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 471: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/471.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u
satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 472: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/472.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 473: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/473.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but, there isonly one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = vDr. Marco A Roque Sol Linear Algebra. Session 9
![Page 474: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/474.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 475: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/475.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though
|A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 476: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/476.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0,
it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 477: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/477.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that
it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 478: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/478.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible
to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 479: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/479.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it
for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 480: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/480.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u
( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 481: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/481.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually,
there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 482: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/482.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are
infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 483: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/483.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .
Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 484: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/484.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now,
Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 485: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/485.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation,
together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 486: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/486.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with
the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 487: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/487.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation
for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 488: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/488.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,
we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 489: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/489.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 490: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/490.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI)
[(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 491: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/491.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 492: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/492.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u =
(A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 493: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/493.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 494: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/494.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u =
0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 495: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/495.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 496: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/496.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector
u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 497: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/497.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u
is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 498: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/498.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as
a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 499: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/499.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 500: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/500.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 501: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/501.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 502: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/502.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason
why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 503: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/503.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system
of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 504: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/504.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear
(algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 505: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/505.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)
equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 506: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/506.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations
presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 507: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/507.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty
is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 508: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/508.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that
the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 509: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/509.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations
are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 510: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/510.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usually
coupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 511: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/511.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 512: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/512.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence,
the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 513: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/513.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations
in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 514: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/514.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system
must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 515: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/515.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved
simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 516: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/516.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.
On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 517: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/517.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary,
if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 518: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/518.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system
is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 519: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/519.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled,
then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 520: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/520.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equation
can be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 521: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/521.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved
independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 522: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/522.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently
of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 523: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/523.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 524: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/524.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming
the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 525: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/525.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system
into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 526: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/526.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent
uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 527: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/527.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem
( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 528: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/528.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which
each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 529: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/529.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation
contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 530: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/530.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only
one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 531: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/531.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable )
corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 532: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/532.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds
to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 533: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/533.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming
the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 534: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/534.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix
A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 535: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/535.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A
intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 536: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/536.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa
diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 537: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/537.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix.
Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 538: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/538.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors
are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 539: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/539.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful
in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 540: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/540.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing
such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 541: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/541.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence, the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 542: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/542.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
U =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 543: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/543.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Let’s assume
that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
U =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 544: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/544.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Let’s assume that the matrix
A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
U =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 545: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/545.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Let’s assume that the matrix A
has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
U =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 546: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/546.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n)
linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
U =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 547: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/547.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
U =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 548: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/548.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
U =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 549: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/549.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and
considering the matrix
U =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 550: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/550.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
U =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 551: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/551.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
U =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 552: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/552.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
we have
AU =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= UD
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 553: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/553.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
we have
AU =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= UD
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 554: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/554.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
we have
AU =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= UD
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 555: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/555.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
we have
AU =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= UD
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 556: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/556.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
we have
AU =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
=
UD
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 557: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/557.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
we have
AU =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= UD
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 558: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/558.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 559: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/559.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D
is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 560: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/560.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 561: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/561.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 562: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/562.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 563: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/563.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose
diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 564: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/564.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements
are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 565: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/565.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues
of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 566: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/566.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A.
From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 567: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/567.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations
we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 568: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/568.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that
it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 569: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/569.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 570: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/570.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 571: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/571.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus,
if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 572: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/572.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and
eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 573: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/573.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors
of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 574: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/574.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A
are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 575: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/575.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known,
A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 576: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/576.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed
into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 577: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/577.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a
diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 578: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/578.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix
by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 579: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/579.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process
shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 580: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/580.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in
theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 581: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/581.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
U−1AU = D ⇐⇒ A = UDU−1
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 582: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/582.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 583: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/583.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process
is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 584: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/584.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as
a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 585: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/585.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.
Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 586: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/586.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively,
we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 587: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/587.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say
that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 588: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/588.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A
is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 589: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/589.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 590: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/590.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If
A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 591: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/591.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A
is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 592: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/592.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian
( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 593: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/593.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ),
then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 594: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/594.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination
of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 595: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/595.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1
isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 596: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/596.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple.
The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 597: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/597.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n)
of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 598: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/598.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A
are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 599: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/599.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known
to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 600: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/600.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to be
mutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 601: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/601.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal,
so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 602: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/602.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us
choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 603: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/603.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them
so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 604: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/604.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that
they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 605: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/605.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are
alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 606: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/606.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized
by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 607: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/607.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1
for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 608: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/608.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i .
It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 609: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/609.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy
verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 610: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/610.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify that
U−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 611: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/611.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗.
In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 612: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/612.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words,
the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 613: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/613.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse
of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 614: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/614.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U
is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 615: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/615.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same
as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 616: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/616.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint
(the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 617: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/617.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose
of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 618: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/618.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its
complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 619: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/619.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 620: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/620.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally,
we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 621: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/621.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note
that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 622: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/622.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A
has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 623: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/623.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer
than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 624: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/624.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n
linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 625: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/625.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors,
then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 626: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/626.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no
matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 627: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/627.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U
such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 628: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/628.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that
U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 629: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/629.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D.
Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 630: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/630.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case,
A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 631: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/631.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A
is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 632: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/632.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not
diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 633: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/633.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian ( A = (A∗)T ), then the determination of U−1 isvery simple. The eigenvectors v(1), ..., v(n) of A are known to bemutually orthogonal, so let us choose them so that they are alsonormalized by < v(i), v(i) >= 1 for each i . It is easy verify thatU−1 = U∗. In other words, the inverse of U is the same as itsadjoint (the transpose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix U such that U−1AU = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 634: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/634.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 635: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/635.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 636: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/636.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 637: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/637.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 638: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/638.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t)
form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 639: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/639.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set
ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 640: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/640.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions
on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 641: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/641.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval
α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 642: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/642.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β.
Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 643: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/643.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 644: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/644.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 645: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/645.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 646: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/646.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns
are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 647: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/647.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors
x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 648: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/648.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t),
is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 649: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/649.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be
afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 650: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/650.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix
for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 651: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/651.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system.
Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 652: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/652.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions
is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 653: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/653.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent
the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 654: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/654.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix
is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 655: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/655.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 656: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/656.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, for example, a fundamental matrix for the system
x′ = Ax =
(1 −11 3
)x
can be formed from the solutions x (1)(t) and x (2)(t):
x(1) =
(1
− 1
)e2t ; x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 657: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/657.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus,
for example, a fundamental matrix for the system
x′ = Ax =
(1 −11 3
)x
can be formed from the solutions x (1)(t) and x (2)(t):
x(1) =
(1
− 1
)e2t ; x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 658: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/658.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, for example,
a fundamental matrix for the system
x′ = Ax =
(1 −11 3
)x
can be formed from the solutions x (1)(t) and x (2)(t):
x(1) =
(1
− 1
)e2t ; x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 659: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/659.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, for example, a fundamental matrix
for the system
x′ = Ax =
(1 −11 3
)x
can be formed from the solutions x (1)(t) and x (2)(t):
x(1) =
(1
− 1
)e2t ; x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 660: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/660.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, for example, a fundamental matrix for the system
x′ = Ax =
(1 −11 3
)x
can be formed from the solutions x (1)(t) and x (2)(t):
x(1) =
(1
− 1
)e2t ; x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 661: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/661.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, for example, a fundamental matrix for the system
x′ = Ax =
(1 −11 3
)x
can be formed from the solutions x (1)(t) and x (2)(t):
x(1) =
(1
− 1
)e2t ; x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 662: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/662.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, for example, a fundamental matrix for the system
x′ = Ax =
(1 −11 3
)x
can be formed from the solutions x (1)(t) and x (2)(t):
x(1) =
(1
− 1
)e2t ; x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 663: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/663.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, for example, a fundamental matrix for the system
x′ = Ax =
(1 −11 3
)x
can be formed
from the solutions x (1)(t) and x (2)(t):
x(1) =
(1
− 1
)e2t ; x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 664: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/664.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, for example, a fundamental matrix for the system
x′ = Ax =
(1 −11 3
)x
can be formed from the solutions
x (1)(t) and x (2)(t):
x(1) =
(1
− 1
)e2t ; x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 665: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/665.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, for example, a fundamental matrix for the system
x′ = Ax =
(1 −11 3
)x
can be formed from the solutions x (1)(t) and
x (2)(t):
x(1) =
(1
− 1
)e2t ; x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 666: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/666.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, for example, a fundamental matrix for the system
x′ = Ax =
(1 −11 3
)x
can be formed from the solutions x (1)(t) and x (2)(t):
x(1) =
(1
− 1
)e2t ; x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 667: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/667.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, for example, a fundamental matrix for the system
x′ = Ax =
(1 −11 3
)x
can be formed from the solutions x (1)(t) and x (2)(t):
x(1) =
(1
− 1
)e2t ;
x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 668: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/668.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, for example, a fundamental matrix for the system
x′ = Ax =
(1 −11 3
)x
can be formed from the solutions x (1)(t) and x (2)(t):
x(1) =
(1
− 1
)e2t ; x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 669: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/669.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, for example, a fundamental matrix for the system
x′ = Ax =
(1 −11 3
)x
can be formed from the solutions x (1)(t) and x (2)(t):
x(1) =
(1
− 1
)e2t ; x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 670: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/670.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, for example, a fundamental matrix for the system
x′ = Ax =
(1 −11 3
)x
can be formed from the solutions x (1)(t) and x (2)(t):
x(1) =
(1
− 1
)e2t ; x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 671: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/671.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, for example, a fundamental matrix for the system
x′ = Ax =
(1 −11 3
)x
can be formed from the solutions x (1)(t) and x (2)(t):
x(1) =
(1
− 1
)e2t ; x(2) =
(1−1
)te2t +
(0−1
)e2t
then
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 672: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/672.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 673: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/673.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 674: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/674.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 675: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/675.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)=
e2t(
1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 676: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/676.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 677: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/677.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)
Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 678: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/678.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column
of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 679: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/679.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t)
is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 680: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/680.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE.
It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 681: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/681.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t)
satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 682: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/682.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 683: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/683.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 684: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/684.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular,
the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 685: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/685.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = I
is called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 686: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/686.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and
can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 687: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/687.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be found
from the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 688: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/688.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation
Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 689: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/689.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) =
Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 690: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/690.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)
Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 691: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/691.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
then
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Iis called the special fundamental matrix and can also be foundfrom the relation Φ(t) = Ψ(t)Ψ−1(0).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 692: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/692.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 693: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/693.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 694: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/694.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 695: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/695.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)
=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 696: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/696.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 697: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/697.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 698: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/698.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) =
Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 699: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/699.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) = Ψ(t)Ψ−1(0) =
e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 700: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/700.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
)
(1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 701: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/701.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 702: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/702.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) =
e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 703: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/703.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)
The latter matrix is also known as the exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 704: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/704.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix
is also known as the exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 705: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/705.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known
as the exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 706: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/706.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the
exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 707: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/707.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, in this case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 708: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/708.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 709: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/709.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 710: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/710.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 711: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/711.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 712: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/712.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 713: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/713.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 714: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/714.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem
for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 715: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/715.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 716: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/716.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 717: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/717.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 718: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/718.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 719: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/719.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Applying the results of already obtained,
we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 720: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/720.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 721: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/721.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 722: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/722.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I.
Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 723: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/723.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat .
let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 724: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/724.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 725: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/725.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat
can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 726: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/726.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 727: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/727.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 728: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/728.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t.
Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 729: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/729.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and
consider the corresponding series
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 730: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/730.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 731: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/731.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 732: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/732.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!=
I + At +A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 733: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/733.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I +
At +A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 734: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/734.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 735: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/735.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+
...+Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 736: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/736.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 737: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/737.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+
...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 738: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/738.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 739: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/739.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix.
It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 740: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/740.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat
each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 741: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/741.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges
for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 742: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/742.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞.
Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 743: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/743.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix,
which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 744: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/744.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 745: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/745.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 746: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/746.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series
term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 747: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/747.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term,
we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 748: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/748.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 749: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/749.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=
∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 750: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/750.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!=
A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 751: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/751.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]=
AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 752: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/752.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 753: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/753.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the special fundamental matrix Φsatisfies the same initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 754: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/754.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the special fundamental matrix Φsatisfies the same initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 755: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/755.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the special fundamental matrix Φsatisfies the same initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 756: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/756.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0
in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the special fundamental matrix Φsatisfies the same initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 757: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/757.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt
we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the special fundamental matrix Φsatisfies the same initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 758: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/758.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the special fundamental matrix Φsatisfies the same initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 759: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/759.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the special fundamental matrix Φsatisfies the same initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 760: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/760.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way,
we have that the special fundamental matrix Φsatisfies the same initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 761: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/761.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that
the special fundamental matrix Φsatisfies the same initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 762: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/762.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the special fundamental matrix Φ
satisfies the same initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 763: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/763.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the special fundamental matrix Φsatisfies the same initial value problem as
eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 764: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/764.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the special fundamental matrix Φsatisfies the same initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 765: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/765.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the special fundamental matrix Φsatisfies the same initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 766: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/766.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the special fundamentalmatrix Φ(t) are the same. Thus, we can write the solution of theinitial value problem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 767: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/767.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Then, by uniqueness of an IVP
(extended to matrix differentialequations), we conclude that eAt and the special fundamentalmatrix Φ(t) are the same. Thus, we can write the solution of theinitial value problem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 768: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/768.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Then, by uniqueness of an IVP (extended to matrix differentialequations),
we conclude that eAt and the special fundamentalmatrix Φ(t) are the same. Thus, we can write the solution of theinitial value problem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 769: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/769.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and
the special fundamentalmatrix Φ(t) are the same. Thus, we can write the solution of theinitial value problem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 770: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/770.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the special fundamentalmatrix
Φ(t) are the same. Thus, we can write the solution of theinitial value problem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 771: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/771.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the special fundamentalmatrix Φ(t)
are the same. Thus, we can write the solution of theinitial value problem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 772: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/772.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the special fundamentalmatrix Φ(t) are the same.
Thus, we can write the solution of theinitial value problem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 773: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/773.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the special fundamentalmatrix Φ(t) are the same. Thus,
we can write the solution of theinitial value problem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 774: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/774.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the special fundamentalmatrix Φ(t) are the same. Thus, we can write
the solution of theinitial value problem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 775: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/775.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the special fundamentalmatrix Φ(t) are the same. Thus, we can write the solution
of theinitial value problem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 776: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/776.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the special fundamentalmatrix Φ(t) are the same. Thus, we can write the solution of theinitial
value problem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 777: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/777.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the special fundamentalmatrix Φ(t) are the same. Thus, we can write the solution of theinitial value problem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 778: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/778.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the special fundamentalmatrix Φ(t) are the same. Thus, we can write the solution of theinitial value problem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 779: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/779.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the special fundamentalmatrix Φ(t) are the same. Thus, we can write the solution of theinitial value problem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 780: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/780.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the special fundamentalmatrix Φ(t) are the same. Thus, we can write the solution of theinitial value problem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 781: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/781.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 782: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/782.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 783: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/783.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An
n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 784: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/784.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix
A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 785: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/785.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A
can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 786: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/786.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A can be diagonalized
only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 787: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/787.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if
it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 788: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/788.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has
a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 789: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/789.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement
of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 790: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/790.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of
n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 791: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/791.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n
linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 792: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/792.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent
eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 793: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/793.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 794: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/794.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors
(because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 795: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/795.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues),
then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 796: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/796.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed
into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 797: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/797.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called
its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 798: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/798.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 799: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/799.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 800: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/800.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form,
J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 801: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/801.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J,
has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 802: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/802.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J, has the eigenvalues
of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 803: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/803.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J, has the eigenvalues of A
on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 804: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/804.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J, has the eigenvalues of A on the
main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 805: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/805.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J, has the eigenvalues of A on the main diagonal,
ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 806: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/806.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions
on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 807: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/807.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal
above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 808: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/808.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and
zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 809: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/809.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 810: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/810.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 811: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/811.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 812: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/812.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 813: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/813.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3
. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 814: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/814.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 815: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/815.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 816: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/816.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 817: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/817.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again
the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 818: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/818.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A
given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 819: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/819.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 820: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/820.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 821: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/821.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 822: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/822.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform
A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 823: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/823.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into
its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 824: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/824.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form,
we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 825: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/825.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct
thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 826: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/826.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix
U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 827: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/827.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U
with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 828: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/828.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector
v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 829: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/829.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v
in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 830: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/830.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and
the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 831: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/831.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector
u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 832: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/832.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 )
in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 833: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/833.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column.
Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 834: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/834.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then
U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 835: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/835.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and
its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 836: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/836.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse
are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 837: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/837.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 838: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/838.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 839: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/839.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)
U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 840: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/840.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 841: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/841.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)
It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 842: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/842.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 843: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/843.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J =
U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 844: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/844.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 845: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/845.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
)
(1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 846: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/846.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
)
(1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 847: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/847.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 848: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/848.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix U with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then U and its inverse are given by
U =
(1 0−1 −1
)U−1 =
(1 0−1 −1
)It follows that
J = U−1AU =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 849: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/849.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Uy where U is given above, produces thesystem
y′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 850: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/850.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Finally,
If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Uy where U is given above, produces thesystem
y′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 851: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/851.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Uy where U is given above, produces thesystem
y′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 852: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/852.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Uy where U is given above, produces thesystem
y′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 853: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/853.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Uy where U is given above, produces thesystem
y′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 854: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/854.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation
x = Uy where U is given above, produces thesystem
y′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 855: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/855.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Uy
where U is given above, produces thesystem
y′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 856: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/856.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Uy where U
is given above, produces thesystem
y′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 857: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/857.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Uy where U is given above,
produces thesystem
y′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 858: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/858.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Uy where U is given above, produces thesystem
y′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 859: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/859.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Uy where U is given above, produces thesystem
y′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 860: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/860.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Uy where U is given above, produces thesystem
y′ = Jy
y ′1 = 2y1 + y2,
y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 861: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/861.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Uy where U is given above, produces thesystem
y′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 862: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/862.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Uy where U is given above, produces thesystem
y′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t ,
y1 = c1te2t + c2e
2t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 863: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/863.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Uy where U is given above, produces thesystem
y′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 864: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/864.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 865: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/865.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus,
two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 866: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/866.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions
of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 867: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/867.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 868: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/868.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 869: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/869.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ;
y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 870: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/870.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 871: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/871.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 872: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/872.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding
fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 873: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/873.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 874: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/874.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 875: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/875.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)
Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 876: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/876.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I,
we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 877: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/877.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also
identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 878: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/878.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt .
To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 879: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/879.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtain
a fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 880: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/880.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix
for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 881: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/881.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system,
we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 882: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/882.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form
theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 883: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/883.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 884: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/884.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) =
UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 885: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/885.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 886: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/886.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)
which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 887: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/887.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is t
he same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 888: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/888.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same
as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 889: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/889.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the
fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
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Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix
that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 891: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/891.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
Complex EigenvaluesRepeated EigenvaluesDiagonalization
Diagonalization.
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = UeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 892: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/892.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 893: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/893.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section,
we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 894: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/894.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A
is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 895: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/895.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrix
with m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 896: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/896.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n.
(This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 897: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/897.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption
is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 898: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/898.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for
convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 899: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/899.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only;
allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 900: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/900.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results
will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 901: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/901.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold
if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 902: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/902.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 903: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/903.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present
a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 904: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/904.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method
for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 905: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/905.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close
A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 906: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/906.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is
to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 907: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/907.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix
of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 908: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/908.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank.
The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 909: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/909.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves
factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 910: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/910.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A
into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 911: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/911.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct
UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 912: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/912.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T ,
where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 913: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/913.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is
an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 914: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/914.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m
orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 915: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/915.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix,
V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 916: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/916.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is an
n × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 917: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/917.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n
orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 918: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/918.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and
Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 919: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/919.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is
an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 920: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/920.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix
whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 921: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/921.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries
are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 922: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/922.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and
whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 923: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/923.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal
elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 924: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/924.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 925: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/925.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Singular Value DecompositionIn this section, we assume throughout that A is an m × n matrixwith m ≥ n. (This assumption is made for convenience only; allthe results will also hold if m < n).
We will present a method for determining how close A is to amatrix of smaller rank. The method involves factoring A into aproduct UΣV T , where U is an m ×m orthogonal matrix, V is ann × n orthogonal matrix, and Σ is an m × n matrix whoseoff-diagonal entries are all 0′s and whose diagonal elements satisfy
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 926: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/926.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Σ =
σ1σ2 . . .
σn
The σ′s determined by this factorization are unique and are calledthe singular values of A. The factorization UΣV T is called thesingular value decomposition of A, or, for short, the SVD of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 927: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/927.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Σ =
σ1σ2 . . .
σn
The σ′s determined by this factorization are unique and are calledthe singular values of A. The factorization UΣV T is called thesingular value decomposition of A, or, for short, the SVD of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 928: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/928.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Σ =
σ1σ2 . . .
σn
The σ′s determined by this factorization are unique and are calledthe singular values of A. The factorization UΣV T is called thesingular value decomposition of A, or, for short, the SVD of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 929: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/929.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Σ =
σ1σ2 . . .
σn
The σ′s determined
by this factorization are unique and are calledthe singular values of A. The factorization UΣV T is called thesingular value decomposition of A, or, for short, the SVD of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 930: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/930.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Σ =
σ1σ2 . . .
σn
The σ′s determined by this factorization
are unique and are calledthe singular values of A. The factorization UΣV T is called thesingular value decomposition of A, or, for short, the SVD of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 931: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/931.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Σ =
σ1σ2 . . .
σn
The σ′s determined by this factorization are unique and
are calledthe singular values of A. The factorization UΣV T is called thesingular value decomposition of A, or, for short, the SVD of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 932: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/932.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Σ =
σ1σ2 . . .
σn
The σ′s determined by this factorization are unique and are called
the singular values of A. The factorization UΣV T is called thesingular value decomposition of A, or, for short, the SVD of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 933: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/933.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Σ =
σ1σ2 . . .
σn
The σ′s determined by this factorization are unique and are calledthe singular values
of A. The factorization UΣV T is called thesingular value decomposition of A, or, for short, the SVD of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 934: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/934.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Σ =
σ1σ2 . . .
σn
The σ′s determined by this factorization are unique and are calledthe singular values of A. The factorization
UΣV T is called thesingular value decomposition of A, or, for short, the SVD of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 935: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/935.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Σ =
σ1σ2 . . .
σn
The σ′s determined by this factorization are unique and are calledthe singular values of A. The factorization UΣV T
is called thesingular value decomposition of A, or, for short, the SVD of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 936: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/936.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Σ =
σ1σ2 . . .
σn
The σ′s determined by this factorization are unique and are calledthe singular values of A. The factorization UΣV T is called
thesingular value decomposition of A, or, for short, the SVD of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 937: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/937.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Σ =
σ1σ2 . . .
σn
The σ′s determined by this factorization are unique and are calledthe singular values of A. The factorization UΣV T is called thesingular value decomposition
of A, or, for short, the SVD of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 938: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/938.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Σ =
σ1σ2 . . .
σn
The σ′s determined by this factorization are unique and are calledthe singular values of A. The factorization UΣV T is called thesingular value decomposition of A, or,
for short, the SVD of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 939: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/939.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Σ =
σ1σ2 . . .
σn
The σ′s determined by this factorization are unique and are calledthe singular values of A. The factorization UΣV T is called thesingular value decomposition of A, or, for short,
the SVD of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 940: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/940.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Σ =
σ1σ2 . . .
σn
The σ′s determined by this factorization are unique and are calledthe singular values of A. The factorization UΣV T is called thesingular value decomposition of A, or, for short, the SVD of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 941: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/941.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 942: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/942.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 943: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/943.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A
is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 944: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/944.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an
m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 945: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/945.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then
A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 946: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/946.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has
a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 947: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/947.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 948: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/948.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 949: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/949.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA
is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 950: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/950.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric
n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 951: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/951.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 952: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/952.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues
of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 953: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/953.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA
are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 954: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/954.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and
it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 955: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/955.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has
an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 956: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/956.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonal
diagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 957: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/957.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix
V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 958: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/958.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 959: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/959.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore,
its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 960: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/960.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues
must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 961: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/961.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be
nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 962: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/962.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 963: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/963.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see
this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 964: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/964.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point,
let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 965: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/965.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ
be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 966: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/966.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue
of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 967: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/967.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and
x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 968: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/968.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x
be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 969: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/969.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector
belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 970: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/970.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ.
It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 971: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/971.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
The SVD Theorem
If A is an m × n matrix, then A has a singular value decomposition
Sketch of the proof
ATA is a symmetric n × n matrix.
The eigenvalues of ATA are all real and it has an orthogonaldiagonalizing matrix V .
Furthermore, its eigenvalues must all be nonnegative.
To see this point, let λ be an eigenvalue of ATA and x be aneigenvector belonging to λ. It follows that
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 972: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/972.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 973: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/973.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 =
xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 974: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/974.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx =
xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 975: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/975.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx =
λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 976: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/976.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 977: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/977.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 978: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/978.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =
||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 979: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/979.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 980: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/980.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume
that the columns of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 981: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/981.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns
of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 982: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/982.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V
have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 983: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/983.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered
sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 984: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/984.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat
the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 985: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/985.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding
eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 986: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/986.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding eigenvalues satisfy
λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 987: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/987.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0.
The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 988: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/988.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values
are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 989: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/989.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 990: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/990.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =
√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 991: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/991.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj ,
j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 992: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/992.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
||Ax||2 = xTATAx = xTλx = λxTx = λ||x||2 ⇒
λ =||Ax||2
||x||2
We may assume that the columns of V have been ordered sothat the corresponding eigenvalues satisfyλ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. The singular values are given by
σj =√λj , j = 1, 2, ..., n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 993: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/993.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 994: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/994.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote
the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 995: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/995.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank
of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 996: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/996.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A.
The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 997: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/997.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix
ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 998: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/998.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA
will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 999: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/999.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also have
rank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1000: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1000.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r .
Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1001: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1001.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA
is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1002: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1002.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric,
its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1003: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1003.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals
the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1004: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1004.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof
nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1005: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1005.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues.
Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1006: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1006.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1007: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1007.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0
σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1008: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1008.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1009: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1009.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now
let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1010: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1010.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and
V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1011: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1011.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1012: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1012.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors
of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1013: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1013.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1
are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1014: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1014.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of
ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1015: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1015.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA
belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1016: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1016.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto
λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1017: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1017.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1018: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1018.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors
of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1019: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1019.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2
are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1020: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1020.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors
of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1021: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1021.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA
belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1022: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1022.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto
λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1023: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1023.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Let r denote the rank of A. The matrix ATA will also haverank r . Since ATA is symmetric, its rank equals the numberof nonzero eigenvalues. Thus,
σ1 ≥ σ2 ≥ · · · ≥ σr > 0 σr+1 = σr+2 = · · · = σn = 0
Now let V1 = (v1, v2, ...., vr , ) and V2 = (vr+1, vr+2, ...., vn, )
The column vectors of V1 are eigenvectors of ATA belongingto λi , i = 1, 2, ..., r .
The column vectors of V2 are eigenvectors of ATA belongingto λj = 0, j = r + 1, r + 2, ..., n.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1024: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1024.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Now let Σ1 be the r × r matrix defined by
Σ1 =
σ1σ2 . . .
σn
The m × n matrix Σ is then given by
Σ =
(Σ1 00 0
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1025: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1025.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Now
let Σ1 be the r × r matrix defined by
Σ1 =
σ1σ2 . . .
σn
The m × n matrix Σ is then given by
Σ =
(Σ1 00 0
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1026: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1026.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Now let Σ1
be the r × r matrix defined by
Σ1 =
σ1σ2 . . .
σn
The m × n matrix Σ is then given by
Σ =
(Σ1 00 0
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1027: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1027.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Now let Σ1 be the r × r matrix
defined by
Σ1 =
σ1σ2 . . .
σn
The m × n matrix Σ is then given by
Σ =
(Σ1 00 0
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1028: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1028.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Now let Σ1 be the r × r matrix defined by
Σ1 =
σ1σ2 . . .
σn
The m × n matrix Σ is then given by
Σ =
(Σ1 00 0
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1029: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1029.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Now let Σ1 be the r × r matrix defined by
Σ1 =
σ1σ2 . . .
σn
The m × n matrix Σ is then given by
Σ =
(Σ1 00 0
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1030: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1030.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Now let Σ1 be the r × r matrix defined by
Σ1 =
σ1σ2 . . .
σn
The m × n matrix Σ is then given by
Σ =
(Σ1 00 0
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1031: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1031.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Now let Σ1 be the r × r matrix defined by
Σ1 =
σ1σ2 . . .
σn
The m × n matrix
Σ is then given by
Σ =
(Σ1 00 0
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1032: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1032.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Now let Σ1 be the r × r matrix defined by
Σ1 =
σ1σ2 . . .
σn
The m × n matrix Σ
is then given by
Σ =
(Σ1 00 0
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1033: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1033.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Now let Σ1 be the r × r matrix defined by
Σ1 =
σ1σ2 . . .
σn
The m × n matrix Σ is then given by
Σ =
(Σ1 00 0
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1034: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1034.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Now let Σ1 be the r × r matrix defined by
Σ1 =
σ1σ2 . . .
σn
The m × n matrix Σ is then given by
Σ =
(Σ1 00 0
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1035: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1035.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Now let Σ1 be the r × r matrix defined by
Σ1 =
σ1σ2 . . .
σn
The m × n matrix Σ is then given by
Σ =
(Σ1 00 0
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1036: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1036.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
To complete the proof, we must show how to construct anm ×m orthogonal matrix U such that
A = UΣV T
AV = UΣ
Comparing the first r columns of each side of the lastequation, we see that
Avi = σivi , i = 1, 2, ..., r
Thus, if we define
ui =1
σiAvi , i = 1, 2, ..., r
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1037: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1037.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
To complete the proof, we must show how to construct anm ×m orthogonal matrix U such that
A = UΣV T
AV = UΣ
Comparing the first r columns of each side of the lastequation, we see that
Avi = σivi , i = 1, 2, ..., r
Thus, if we define
ui =1
σiAvi , i = 1, 2, ..., r
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1038: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1038.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
To complete the proof, we must show how to construct anm ×m orthogonal matrix U such that
A = UΣV T
AV = UΣ
Comparing the first r columns of each side of the lastequation, we see that
Avi = σivi , i = 1, 2, ..., r
Thus, if we define
ui =1
σiAvi , i = 1, 2, ..., r
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1039: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1039.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
To complete the proof, we must show how to construct anm ×m orthogonal matrix U such that
A = UΣV T
AV = UΣ
Comparing the first r columns of each side of the lastequation, we see that
Avi = σivi , i = 1, 2, ..., r
Thus, if we define
ui =1
σiAvi , i = 1, 2, ..., r
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1040: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1040.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
To complete the proof, we must show how to construct anm ×m orthogonal matrix U such that
A = UΣV T
AV = UΣ
Comparing the first r columns of each side of the lastequation, we see that
Avi = σivi , i = 1, 2, ..., r
Thus, if we define
ui =1
σiAvi , i = 1, 2, ..., r
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1041: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1041.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
To complete the proof, we must show how to construct anm ×m orthogonal matrix U such that
A = UΣV T
AV = UΣ
Comparing the first r columns of each side of the lastequation, we see that
Avi = σivi , i = 1, 2, ..., r
Thus, if we define
ui =1
σiAvi , i = 1, 2, ..., r
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1042: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1042.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
To complete the proof, we must show how to construct anm ×m orthogonal matrix U such that
A = UΣV T
AV = UΣ
Comparing the first r columns of each side of the lastequation, we see that
Avi = σivi , i = 1, 2, ..., r
Thus, if we define
ui =1
σiAvi , i = 1, 2, ..., r
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1043: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1043.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
To complete the proof, we must show how to construct anm ×m orthogonal matrix U such that
A = UΣV T
AV = UΣ
Comparing the first r columns of each side of the lastequation, we see that
Avi = σivi , i = 1, 2, ..., r
Thus, if we define
ui =1
σiAvi , i = 1, 2, ..., r
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1044: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1044.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1045: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1045.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1046: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1046.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1047: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1047.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1048: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1048.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1049: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1049.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors
of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1050: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1050.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1
form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1051: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1051.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set.
Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1052: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1052.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,
form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1053: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1053.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal
basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1054: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1054.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for
R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1055: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1055.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A).
The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1056: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1056.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector space
R(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1057: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1057.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT )
has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1058: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1058.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension
m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1059: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1059.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r .
Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1060: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1060.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let
{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1061: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1061.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un}
be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1062: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1062.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis
for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1063: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1063.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1064: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1064.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
and
U1 = (u1,u2, ...,ur )
then it follows that
AV1 = U1Σ1
The column vectors of U1 form an orthonormal set. Thus,form an orthonormal basis for R(A). The vector spaceR(A)⊥ = N(AT ) has dimension m − r . Let{ur+1,ur+2, · · · ,un} be an orthonormal basis for N(AT ) andset
U2 = (ur+1,ur+2, ...,un)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1065: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1065.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Setthe m × n matrix U by
U = (U1,U2)
The the matrices U,Σ, and V satisfy
A = UΣV T �
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1066: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1066.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Setthe m × n matrix U by
U = (U1,U2)
The the matrices U,Σ, and V satisfy
A = UΣV T �
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1067: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1067.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Setthe m × n matrix U by
U = (U1,U2)
The the matrices U,Σ, and V satisfy
A = UΣV T �
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1068: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1068.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Setthe m × n matrix U by
U = (U1,U2)
The the matrices U,Σ, and V satisfy
A = UΣV T �
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1069: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1069.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Setthe m × n matrix U by
U = (U1,U2)
The the matrices U,Σ, and V satisfy
A =
UΣV T �
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1070: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1070.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Setthe m × n matrix U by
U = (U1,U2)
The the matrices U,Σ, and V satisfy
A = U
ΣV T �
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1071: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1071.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Setthe m × n matrix U by
U = (U1,U2)
The the matrices U,Σ, and V satisfy
A = UΣ
V T �
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1072: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1072.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Setthe m × n matrix U by
U = (U1,U2)
The the matrices U,Σ, and V satisfy
A = UΣV T
�
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1073: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1073.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Setthe m × n matrix U by
U = (U1,U2)
The the matrices U,Σ, and V satisfy
A = UΣV T �
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1074: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1074.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Setthe m × n matrix U by
U = (U1,U2)
The the matrices U,Σ, and V satisfy
A = UΣV T �
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1075: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1075.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Observations
Let A be an m × n matrix with a singular value decompositionA = UΣV T .
The singular values σ1, ..., σn of A are unique; however, thematrices U and V are not unique.
Since V diagonalizes ATA, it follows that the v′js are
eigenvectors of ATA.
Since AAT = UΣΣTUT , it follows that U diagonalizes AAT
and that the u′js are eigenvectors of AAT .
The v′js are called the right singular vectors of A, and the u′jsare called the left singular vectors of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1076: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1076.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Observations
Let A be an m × n matrix with a singular value decompositionA = UΣV T .
The singular values σ1, ..., σn of A are unique; however, thematrices U and V are not unique.
Since V diagonalizes ATA, it follows that the v′js are
eigenvectors of ATA.
Since AAT = UΣΣTUT , it follows that U diagonalizes AAT
and that the u′js are eigenvectors of AAT .
The v′js are called the right singular vectors of A, and the u′jsare called the left singular vectors of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1077: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1077.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Observations
Let A be an m × n matrix with a singular value decompositionA = UΣV T .
The singular values σ1, ..., σn of A are unique; however, thematrices U and V are not unique.
Since V diagonalizes ATA, it follows that the v′js are
eigenvectors of ATA.
Since AAT = UΣΣTUT , it follows that U diagonalizes AAT
and that the u′js are eigenvectors of AAT .
The v′js are called the right singular vectors of A, and the u′jsare called the left singular vectors of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1078: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1078.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Observations
Let A be an m × n matrix with a singular value decompositionA = UΣV T .
The singular values σ1, ..., σn of A are unique; however, thematrices U and V are not unique.
Since V diagonalizes ATA, it follows that the v′js are
eigenvectors of ATA.
Since AAT = UΣΣTUT , it follows that U diagonalizes AAT
and that the u′js are eigenvectors of AAT .
The v′js are called the right singular vectors of A, and the u′jsare called the left singular vectors of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1079: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1079.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Observations
Let A be an m × n matrix with a singular value decompositionA = UΣV T .
The singular values σ1, ..., σn of A are unique; however, thematrices U and V are not unique.
Since V diagonalizes ATA, it follows that the v′js are
eigenvectors of ATA.
Since AAT = UΣΣTUT , it follows that U diagonalizes AAT
and that the u′js are eigenvectors of AAT .
The v′js are called the right singular vectors of A, and the u′jsare called the left singular vectors of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1080: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1080.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Observations
Let A be an m × n matrix with a singular value decompositionA = UΣV T .
The singular values σ1, ..., σn of A are unique; however, thematrices U and V are not unique.
Since V diagonalizes ATA, it follows that the v′js are
eigenvectors of ATA.
Since AAT = UΣΣTUT , it follows that U diagonalizes AAT
and that the u′js are eigenvectors of AAT .
The v′js are called the right singular vectors of A, and the u′jsare called the left singular vectors of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1081: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1081.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Observations
Let A be an m × n matrix with a singular value decompositionA = UΣV T .
The singular values σ1, ..., σn of A are unique; however, thematrices U and V are not unique.
Since V diagonalizes ATA, it follows that the v′js are
eigenvectors of ATA.
Since AAT = UΣΣTUT , it follows that U diagonalizes AAT
and that the u′js are eigenvectors of AAT .
The v′js are called the right singular vectors of A, and the u′jsare called the left singular vectors of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1082: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1082.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Observations
Let A be an m × n matrix with a singular value decompositionA = UΣV T .
The singular values σ1, ..., σn of A are unique; however, thematrices U and V are not unique.
Since V diagonalizes ATA, it follows that the v′js are
eigenvectors of ATA.
Since AAT = UΣΣTUT , it follows that U diagonalizes AAT
and that the u′js are eigenvectors of AAT .
The v′js are called the right singular vectors of A, and the u′jsare called the left singular vectors of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1083: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1083.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Observations
Let A be an m × n matrix with a singular value decompositionA = UΣV T .
The singular values σ1, ..., σn of A are unique; however, thematrices U and V are not unique.
Since V diagonalizes ATA, it follows that the v′js are
eigenvectors of ATA.
Since AAT = UΣΣTUT , it follows that U diagonalizes AAT
and that the u′js are eigenvectors of AAT .
The v′js are called the right singular vectors of A, and the u′jsare called the left singular vectors of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1084: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1084.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
Observations
Let A be an m × n matrix with a singular value decompositionA = UΣV T .
The singular values σ1, ..., σn of A are unique; however, thematrices U and V are not unique.
Since V diagonalizes ATA, it follows that the v′js are
eigenvectors of ATA.
Since AAT = UΣΣTUT , it follows that U diagonalizes AAT
and that the u′js are eigenvectors of AAT .
The v′js are called the right singular vectors of A, and the u′jsare called the left singular vectors of A.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1085: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1085.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
If A has rank r , then
(i) v1, v2, ..., vr form an orthonormal basis for R(AT ).
(ii) vr+1, vr+2, ..., vn form an orthonormal basis for N(A).
(iii) u1,u2, ...,ur form an orthonormal basis for R(A).
(iv) ur+1,ur+2, ...,ur+n form an orthonormal basis for N(AT )
The rank of the matrix A is equal to the number of itsnonzero singular values (where singular values are countedaccording to multiplicity).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1086: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1086.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
If A has rank r , then
(i) v1, v2, ..., vr form an orthonormal basis for R(AT ).
(ii) vr+1, vr+2, ..., vn form an orthonormal basis for N(A).
(iii) u1,u2, ...,ur form an orthonormal basis for R(A).
(iv) ur+1,ur+2, ...,ur+n form an orthonormal basis for N(AT )
The rank of the matrix A is equal to the number of itsnonzero singular values (where singular values are countedaccording to multiplicity).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1087: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1087.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
If A has rank r , then
(i) v1, v2, ..., vr form an orthonormal basis for R(AT ).
(ii) vr+1, vr+2, ..., vn form an orthonormal basis for N(A).
(iii) u1,u2, ...,ur form an orthonormal basis for R(A).
(iv) ur+1,ur+2, ...,ur+n form an orthonormal basis for N(AT )
The rank of the matrix A is equal to the number of itsnonzero singular values (where singular values are countedaccording to multiplicity).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1088: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1088.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
If A has rank r , then
(i) v1, v2, ..., vr form an orthonormal basis for R(AT ).
(ii) vr+1, vr+2, ..., vn form an orthonormal basis for N(A).
(iii) u1,u2, ...,ur form an orthonormal basis for R(A).
(iv) ur+1,ur+2, ...,ur+n form an orthonormal basis for N(AT )
The rank of the matrix A is equal to the number of itsnonzero singular values (where singular values are countedaccording to multiplicity).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1089: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1089.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
If A has rank r , then
(i) v1, v2, ..., vr form an orthonormal basis for R(AT ).
(ii) vr+1, vr+2, ..., vn form an orthonormal basis for N(A).
(iii) u1,u2, ...,ur form an orthonormal basis for R(A).
(iv) ur+1,ur+2, ...,ur+n form an orthonormal basis for N(AT )
The rank of the matrix A is equal to the number of itsnonzero singular values (where singular values are countedaccording to multiplicity).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1090: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1090.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
If A has rank r , then
(i) v1, v2, ..., vr form an orthonormal basis for R(AT ).
(ii) vr+1, vr+2, ..., vn form an orthonormal basis for N(A).
(iii) u1,u2, ...,ur form an orthonormal basis for R(A).
(iv) ur+1,ur+2, ...,ur+n form an orthonormal basis for N(AT )
The rank of the matrix A is equal to the number of itsnonzero singular values (where singular values are countedaccording to multiplicity).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1091: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1091.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
If A has rank r , then
(i) v1, v2, ..., vr form an orthonormal basis for R(AT ).
(ii) vr+1, vr+2, ..., vn form an orthonormal basis for N(A).
(iii) u1,u2, ...,ur form an orthonormal basis for R(A).
(iv) ur+1,ur+2, ...,ur+n form an orthonormal basis for N(AT )
The rank of the matrix A is equal to the number of itsnonzero singular values (where singular values are countedaccording to multiplicity).
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1092: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1092.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
In the case that A has rank r < n, if we set
V1 = (v1, v2, ..., vr ) U1 = (u1,u2, ...,ur )and define Σ1 as before, then
A = U1Σ1VT1
This factorization, is called the compact form of thesingular value decomposition of A. This form is useful inmany applications
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1093: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1093.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
In the case that A has rank r < n, if we set
V1 = (v1, v2, ..., vr ) U1 = (u1,u2, ...,ur )and define Σ1 as before, then
A = U1Σ1VT1
This factorization, is called the compact form of thesingular value decomposition of A. This form is useful inmany applications
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1094: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1094.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
In the case that A has rank r < n, if we set
V1 = (v1, v2, ..., vr )
U1 = (u1,u2, ...,ur )and define Σ1 as before, then
A = U1Σ1VT1
This factorization, is called the compact form of thesingular value decomposition of A. This form is useful inmany applications
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1095: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1095.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
In the case that A has rank r < n, if we set
V1 = (v1, v2, ..., vr ) U1 = (u1,u2, ...,ur )and define Σ1 as before, then
A = U1Σ1VT1
This factorization, is called the compact form of thesingular value decomposition of A. This form is useful inmany applications
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1096: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1096.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
In the case that A has rank r < n, if we set
V1 = (v1, v2, ..., vr ) U1 = (u1,u2, ...,ur )and define Σ1 as before, then
A =
U1Σ1VT1
This factorization, is called the compact form of thesingular value decomposition of A. This form is useful inmany applications
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1097: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1097.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
In the case that A has rank r < n, if we set
V1 = (v1, v2, ..., vr ) U1 = (u1,u2, ...,ur )and define Σ1 as before, then
A = U1
Σ1VT1
This factorization, is called the compact form of thesingular value decomposition of A. This form is useful inmany applications
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1098: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1098.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
In the case that A has rank r < n, if we set
V1 = (v1, v2, ..., vr ) U1 = (u1,u2, ...,ur )and define Σ1 as before, then
A = U1Σ1
V T1
This factorization, is called the compact form of thesingular value decomposition of A. This form is useful inmany applications
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1099: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1099.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
In the case that A has rank r < n, if we set
V1 = (v1, v2, ..., vr ) U1 = (u1,u2, ...,ur )and define Σ1 as before, then
A = U1Σ1VT1
This factorization, is called the compact form of thesingular value decomposition of A. This form is useful inmany applications
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1100: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1100.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
In the case that A has rank r < n, if we set
V1 = (v1, v2, ..., vr ) U1 = (u1,u2, ...,ur )and define Σ1 as before, then
A = U1Σ1VT1
This factorization,
is called the compact form of thesingular value decomposition of A. This form is useful inmany applications
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1101: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1101.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
In the case that A has rank r < n, if we set
V1 = (v1, v2, ..., vr ) U1 = (u1,u2, ...,ur )and define Σ1 as before, then
A = U1Σ1VT1
This factorization, is called
the compact form of thesingular value decomposition of A. This form is useful inmany applications
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1102: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1102.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
In the case that A has rank r < n, if we set
V1 = (v1, v2, ..., vr ) U1 = (u1,u2, ...,ur )and define Σ1 as before, then
A = U1Σ1VT1
This factorization, is called the compact form of thesingular value decomposition
of A. This form is useful inmany applications
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1103: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1103.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
In the case that A has rank r < n, if we set
V1 = (v1, v2, ..., vr ) U1 = (u1,u2, ...,ur )and define Σ1 as before, then
A = U1Σ1VT1
This factorization, is called the compact form of thesingular value decomposition of A. This form
is useful inmany applications
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1104: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1104.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
In the case that A has rank r < n, if we set
V1 = (v1, v2, ..., vr ) U1 = (u1,u2, ...,ur )and define Σ1 as before, then
A = U1Σ1VT1
This factorization, is called the compact form of thesingular value decomposition of A. This form is useful
inmany applications
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1105: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1105.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Introduction
In the case that A has rank r < n, if we set
V1 = (v1, v2, ..., vr ) U1 = (u1,u2, ...,ur )and define Σ1 as before, then
A = U1Σ1VT1
This factorization, is called the compact form of thesingular value decomposition of A. This form is useful inmany applications
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1106: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1106.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.7let
A =
1 11 10 0
Compute the singular values and the singular value decompositionof A
Solution
The matrix
ATA =
(2 22 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1107: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1107.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.7
let
A =
1 11 10 0
Compute the singular values and the singular value decompositionof A
Solution
The matrix
ATA =
(2 22 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1108: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1108.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.7let
A =
1 11 10 0
Compute the singular values and the singular value decompositionof A
Solution
The matrix
ATA =
(2 22 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1109: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1109.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.7let
A =
1 11 10 0
Compute the singular values and the singular value decompositionof A
Solution
The matrix
ATA =
(2 22 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1110: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1110.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.7let
A =
1 11 10 0
Compute the singular values and the singular value decompositionof A
Solution
The matrix
ATA =
(2 22 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1111: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1111.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.7let
A =
1 11 10 0
Compute
the singular values and the singular value decompositionof A
Solution
The matrix
ATA =
(2 22 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1112: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1112.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.7let
A =
1 11 10 0
Compute the singular values and
the singular value decompositionof A
Solution
The matrix
ATA =
(2 22 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1113: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1113.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.7let
A =
1 11 10 0
Compute the singular values and the singular value decomposition
of A
Solution
The matrix
ATA =
(2 22 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1114: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1114.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.7let
A =
1 11 10 0
Compute the singular values and the singular value decompositionof A
Solution
The matrix
ATA =
(2 22 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1115: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1115.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.7let
A =
1 11 10 0
Compute the singular values and the singular value decompositionof A
Solution
The matrix
ATA =
(2 22 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1116: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1116.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.7let
A =
1 11 10 0
Compute the singular values and the singular value decompositionof A
Solution
The matrix
ATA =
(2 22 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1117: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1117.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.7let
A =
1 11 10 0
Compute the singular values and the singular value decompositionof A
Solution
The matrix
ATA =
(2 22 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1118: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1118.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.7let
A =
1 11 10 0
Compute the singular values and the singular value decompositionof A
Solution
The matrix
ATA =
(2 22 2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1119: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1119.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1120: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1120.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues
λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1121: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1121.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.
Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1122: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1122.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently,
the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1123: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1123.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues
of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1124: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1124.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are
σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1125: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1125.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and
σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1126: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1126.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0
The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1127: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1127.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue
λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1128: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1128.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1
haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1129: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1129.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors
of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1130: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1130.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form
α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1131: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1131.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and
σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1132: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1132.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors
of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1133: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1133.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform
β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1134: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1134.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T .
Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1135: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1135.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore,
the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1136: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1136.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1137: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1137.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =
1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1138: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1138.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1139: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1139.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 4 and λ2 = 0.Consequently, the singularvalues of A are σ1 = 2 and σ2 = 0 The eigenvalue λ1 haseigenvectors of the form α(1, 1)T , and σ2 has eigenvectors of theform β(1, 1)T . Therefore, the orthogonal matrix
V =1√2
(1 11 −1
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1140: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1140.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1141: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1141.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes
ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1142: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1142.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA.
From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1143: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1143.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before,
it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1144: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1144.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1145: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1145.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1146: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1146.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1147: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1147.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1148: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1148.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1149: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1149.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1150: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1150.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1151: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1151.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining
column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1152: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1152.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors
of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1153: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1153.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U
must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1154: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1154.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form
an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1155: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1155.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormal
basis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1156: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1156.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for
N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1157: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1157.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ).
We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1158: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1158.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute
a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1159: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1159.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis
{x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1160: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1160.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3}
for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1161: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1161.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT )
inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1162: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1162.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1163: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1163.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T ,
x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1164: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1164.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1165: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1165.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since
these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1166: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1166.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors
are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1167: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1167.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal,
it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1168: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1168.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary
touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1169: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1169.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse
the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1170: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1170.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process
to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1171: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1171.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain
an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1172: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1172.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
diagonalizes ATA. From what we discussed before, it follows that
u1 =1
σ1Av1 =
1
2
1 11 10 0
(1/√
2
1/√
2
)=
1/√
2
1/√
20
The remaining column vectors of U must form an orthonormalbasis for N(AT ). We can compute a basis {x2, x3} for N(AT ) inthe usual way.
x2 = (1,−1, 0)T , x3 = (0, 0, 1)T
Since these vectors are already orthogonal, it is not necessary touse the Gram-Schmidt process to obtain an orthonormal basis.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1173: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1173.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only set
u2 =1
||x2||x2 = (
1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 = (0, 0, 1)T
It then follows that
A = UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 0
0 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1174: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1174.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only
set
u2 =1
||x2||x2 = (
1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 = (0, 0, 1)T
It then follows that
A = UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 0
0 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1175: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1175.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only set
u2 =1
||x2||x2 = (
1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 = (0, 0, 1)T
It then follows that
A = UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 0
0 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1176: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1176.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only set
u2 =
1
||x2||x2 = (
1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 = (0, 0, 1)T
It then follows that
A = UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 0
0 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1177: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1177.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only set
u2 =1
||x2||x2 =
(1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 = (0, 0, 1)T
It then follows that
A = UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 0
0 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1178: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1178.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only set
u2 =1
||x2||x2 = (
1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 = (0, 0, 1)T
It then follows that
A = UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 0
0 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1179: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1179.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only set
u2 =1
||x2||x2 = (
1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 =
(0, 0, 1)T
It then follows that
A = UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 0
0 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1180: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1180.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only set
u2 =1
||x2||x2 = (
1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 = (0, 0, 1)T
It then follows that
A = UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 0
0 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1181: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1181.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only set
u2 =1
||x2||x2 = (
1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 = (0, 0, 1)T
It then
follows that
A = UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 0
0 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1182: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1182.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only set
u2 =1
||x2||x2 = (
1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 = (0, 0, 1)T
It then follows that
A = UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 0
0 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1183: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1183.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only set
u2 =1
||x2||x2 = (
1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 = (0, 0, 1)T
It then follows that
A =
UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 0
0 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1184: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1184.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only set
u2 =1
||x2||x2 = (
1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 = (0, 0, 1)T
It then follows that
A = UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 0
0 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1185: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1185.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only set
u2 =1
||x2||x2 = (
1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 = (0, 0, 1)T
It then follows that
A = UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 00 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1186: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1186.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only set
u2 =1
||x2||x2 = (
1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 = (0, 0, 1)T
It then follows that
A = UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 0
0 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1187: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1187.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only set
u2 =1
||x2||x2 = (
1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 = (0, 0, 1)T
It then follows that
A = UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 0
0 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1188: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1188.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
We need only set
u2 =1
||x2||x2 = (
1√2,− 1√
2, 0)T , u3 =
1
||x3||x3 = (0, 0, 1)T
It then follows that
A = UΣV T =
1√2
1√2
01√2− 1√
20
0 0 1
2 0
0 00 0
(1√2
1√2
1√2− 1√
2
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1189: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1189.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.8Let
A =
(3 2 22 3 −2
)
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
(17 88 17
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1190: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1190.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.8
Let
A =
(3 2 22 3 −2
)
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
(17 88 17
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1191: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1191.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.8Let
A =
(3 2 22 3 −2
)
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
(17 88 17
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1192: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1192.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.8Let
A =
(3 2 22 3 −2
)
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
(17 88 17
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1193: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1193.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.8Let
A =
(3 2 22 3 −2
)
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
(17 88 17
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1194: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1194.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.8Let
A =
(3 2 22 3 −2
)
Compute
the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
(17 88 17
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1195: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1195.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.8Let
A =
(3 2 22 3 −2
)
Compute the singular values and
the singular value decompositionof A.
Solution
The matrix
AAT =
(17 88 17
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1196: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1196.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.8Let
A =
(3 2 22 3 −2
)
Compute the singular values and the singular value decomposition
of A.
Solution
The matrix
AAT =
(17 88 17
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1197: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1197.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.8Let
A =
(3 2 22 3 −2
)
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
(17 88 17
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1198: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1198.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.8Let
A =
(3 2 22 3 −2
)
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
(17 88 17
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1199: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1199.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.8Let
A =
(3 2 22 3 −2
)
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
(17 88 17
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1200: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1200.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.8Let
A =
(3 2 22 3 −2
)
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
(17 88 17
)
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1201: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1201.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.8Let
A =
(3 2 22 3 −2
)
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
(17 88 17
)Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1202: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1202.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1203: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1203.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues
λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1204: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1204.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9.
Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1205: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1205.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently,
the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1206: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1206.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues
of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1207: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1207.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are
σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1208: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1208.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and
σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1209: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1209.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and
σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1210: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1210.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors
(the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1211: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1211.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1212: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1212.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now
we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1213: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1213.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find
the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1214: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1214.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors
(the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1215: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1215.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )
byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1216: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1216.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding
an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1217: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1217.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors
of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1218: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1218.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1219: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1219.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible
to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1220: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1220.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding
the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1221: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1221.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors
(columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1222: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1222.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU)
instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1223: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1223.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead.
The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1224: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1224.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues
of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1225: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1225.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA
are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1226: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1226.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and
sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1227: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1227.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric
we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1228: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1228.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that
the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1229: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1229.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1230: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1230.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25,
we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1231: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1231.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1232: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1232.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1233: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1233.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
has eigenvalues λ1 = 25 and λ2 = 9. Consequently, the singularvalues of A are σ1 = 5 and σ2 = 3, and from here we can find theleft singular vectors (the columns of U ) but we’ll do it in adifferent way at the end.
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA. It is alsopossible to proceed by finding the left singular vectors (columns ofU) instead. The eigenvalues of ATA are 25, 9, and 0, and sinceATA is symmetric we know that the eigenvectors will beorthogonal.
For λ = 25, we have
ATA− 25I =
−12 12 212 −12 −22 −2 −17
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1234: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1234.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 −1 00 0 10 0 0
A unit-length vector in the kernel of that matrix
v1 =1√2
110
For λ = 9, we have
ATA− 9I =
4 12 212 4 −22 −2 −1
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1235: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1235.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 −1 00 0 10 0 0
A unit-length vector in the kernel of that matrix
v1 =1√2
110
For λ = 9, we have
ATA− 9I =
4 12 212 4 −22 −2 −1
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1236: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1236.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 −1 00 0 10 0 0
A unit-length vector
in the kernel of that matrix
v1 =1√2
110
For λ = 9, we have
ATA− 9I =
4 12 212 4 −22 −2 −1
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1237: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1237.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 −1 00 0 10 0 0
A unit-length vector in the kernel of that matrix
v1 =1√2
110
For λ = 9, we have
ATA− 9I =
4 12 212 4 −22 −2 −1
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1238: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1238.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 −1 00 0 10 0 0
A unit-length vector in the kernel of that matrix
v1 =
1√2
110
For λ = 9, we have
ATA− 9I =
4 12 212 4 −22 −2 −1
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1239: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1239.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 −1 00 0 10 0 0
A unit-length vector in the kernel of that matrix
v1 =1√2
110
For λ = 9, we have
ATA− 9I =
4 12 212 4 −22 −2 −1
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1240: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1240.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 −1 00 0 10 0 0
A unit-length vector in the kernel of that matrix
v1 =1√2
110
For λ = 9, we have
ATA− 9I =
4 12 212 4 −22 −2 −1
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1241: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1241.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 −1 00 0 10 0 0
A unit-length vector in the kernel of that matrix
v1 =1√2
110
For λ = 9,
we have
ATA− 9I =
4 12 212 4 −22 −2 −1
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1242: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1242.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 −1 00 0 10 0 0
A unit-length vector in the kernel of that matrix
v1 =1√2
110
For λ = 9, we have
ATA− 9I =
4 12 212 4 −22 −2 −1
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1243: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1243.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 −1 00 0 10 0 0
A unit-length vector in the kernel of that matrix
v1 =1√2
110
For λ = 9, we have
ATA− 9I =
4 12 212 4 −22 −2 −1
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1244: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1244.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 −1 00 0 10 0 0
A unit-length vector in the kernel of that matrix
v1 =1√2
110
For λ = 9, we have
ATA− 9I =
4 12 212 4 −22 −2 −1
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1245: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1245.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1246: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1246.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1247: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1247.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector
in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1248: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1248.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1249: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1249.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =
1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1250: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1250.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1251: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1251.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1252: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1252.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector,
we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1253: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1253.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute
the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1254: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1254.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA or
find a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1255: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1255.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector
perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1256: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1256.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2.
To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1257: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1257.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicular
to v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1258: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1258.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c),
must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1259: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1259.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that
−a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1260: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1260.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and
to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1261: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1261.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular
to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1262: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1262.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2,
vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1263: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1263.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c),
must satisfy that2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1264: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1264.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that
2√18
+ 4c√18
= 0.
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1265: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1265.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
which row-reduces to 1 0 −1/40 1 1/40 0 0
A unit-length vector in the kernel of that matrix
v2 =1√18
1−14
For the last eigenvector, we could compute the kernel of ATA orfind a unit vector perpendicular to v1 and v2. To be perpendicularto v1, vT3 = (a, b, c), must satisfy that −a = b and to beperpendicular to v2, vT3 = (a,−a, c), must satisfy that2√18
+ 4c√18
= 0.Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1266: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1266.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
So
v2 =1√18
a−a−a/2
and for it to be unit-length we need a = 2/3
v3 =1√18
2/3−2/3−1/3
So at this point we know that
A = UΣV T = U
(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1267: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1267.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
So
v2 =
1√18
a−a−a/2
and for it to be unit-length we need a = 2/3
v3 =1√18
2/3−2/3−1/3
So at this point we know that
A = UΣV T = U
(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1268: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1268.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
So
v2 =1√18
a−a−a/2
and for it to be unit-length we need a = 2/3
v3 =1√18
2/3−2/3−1/3
So at this point we know that
A = UΣV T = U
(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1269: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1269.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
So
v2 =1√18
a−a−a/2
and for it to be unit-length we need a = 2/3
v3 =1√18
2/3−2/3−1/3
So at this point we know that
A = UΣV T = U
(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1270: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1270.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
So
v2 =1√18
a−a−a/2
and for it
to be unit-length we need a = 2/3
v3 =1√18
2/3−2/3−1/3
So at this point we know that
A = UΣV T = U
(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1271: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1271.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
So
v2 =1√18
a−a−a/2
and for it to be unit-length
we need a = 2/3
v3 =1√18
2/3−2/3−1/3
So at this point we know that
A = UΣV T = U
(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1272: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1272.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
So
v2 =1√18
a−a−a/2
and for it to be unit-length we need a = 2/3
v3 =1√18
2/3−2/3−1/3
So at this point we know that
A = UΣV T = U
(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1273: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1273.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
So
v2 =1√18
a−a−a/2
and for it to be unit-length we need a = 2/3
v3 =
1√18
2/3−2/3−1/3
So at this point we know that
A = UΣV T = U
(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1274: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1274.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
So
v2 =1√18
a−a−a/2
and for it to be unit-length we need a = 2/3
v3 =1√18
2/3−2/3−1/3
So at this point we know that
A = UΣV T = U
(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1275: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1275.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
So
v2 =1√18
a−a−a/2
and for it to be unit-length we need a = 2/3
v3 =1√18
2/3−2/3−1/3
So at this point we know that
A = UΣV T = U
(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1276: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1276.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
So
v2 =1√18
a−a−a/2
and for it to be unit-length we need a = 2/3
v3 =1√18
2/3−2/3−1/3
So at this point
we know that
A = UΣV T = U
(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1277: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1277.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
So
v2 =1√18
a−a−a/2
and for it to be unit-length we need a = 2/3
v3 =1√18
2/3−2/3−1/3
So at this point we know that
A = UΣV T = U
(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1278: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1278.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
So
v2 =1√18
a−a−a/2
and for it to be unit-length we need a = 2/3
v3 =1√18
2/3−2/3−1/3
So at this point we know that
A = UΣV T = U
(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1279: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1279.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Finally, we can compute U by the formula σui = Avi orui = 1
σAvi . This gives
U =
(1/√
2 1/√
2
1/√
2 −1/√
2
)So the full SVD is:
A = UΣV T =(1/√
2 1/√
2
1/√
2 −1/√
2
)(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1280: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1280.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Finally,
we can compute U by the formula σui = Avi orui = 1
σAvi . This gives
U =
(1/√
2 1/√
2
1/√
2 −1/√
2
)So the full SVD is:
A = UΣV T =(1/√
2 1/√
2
1/√
2 −1/√
2
)(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1281: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1281.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Finally, we can compute U
by the formula σui = Avi orui = 1
σAvi . This gives
U =
(1/√
2 1/√
2
1/√
2 −1/√
2
)So the full SVD is:
A = UΣV T =(1/√
2 1/√
2
1/√
2 −1/√
2
)(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1282: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1282.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Finally, we can compute U by the formula σui = Avi or
ui = 1σAvi . This gives
U =
(1/√
2 1/√
2
1/√
2 −1/√
2
)So the full SVD is:
A = UΣV T =(1/√
2 1/√
2
1/√
2 −1/√
2
)(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1283: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1283.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Finally, we can compute U by the formula σui = Avi orui = 1
σAvi .
This gives
U =
(1/√
2 1/√
2
1/√
2 −1/√
2
)So the full SVD is:
A = UΣV T =(1/√
2 1/√
2
1/√
2 −1/√
2
)(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1284: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1284.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Finally, we can compute U by the formula σui = Avi orui = 1
σAvi . This gives
U =
(1/√
2 1/√
2
1/√
2 −1/√
2
)So the full SVD is:
A = UΣV T =(1/√
2 1/√
2
1/√
2 −1/√
2
)(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1285: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1285.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Finally, we can compute U by the formula σui = Avi orui = 1
σAvi . This gives
U =
(1/√
2 1/√
2
1/√
2 −1/√
2
)
So the full SVD is:
A = UΣV T =(1/√
2 1/√
2
1/√
2 −1/√
2
)(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1286: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1286.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Finally, we can compute U by the formula σui = Avi orui = 1
σAvi . This gives
U =
(1/√
2 1/√
2
1/√
2 −1/√
2
)So
the full SVD is:
A = UΣV T =(1/√
2 1/√
2
1/√
2 −1/√
2
)(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1287: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1287.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Finally, we can compute U by the formula σui = Avi orui = 1
σAvi . This gives
U =
(1/√
2 1/√
2
1/√
2 −1/√
2
)So the full SVD is:
A = UΣV T =(1/√
2 1/√
2
1/√
2 −1/√
2
)(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1288: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1288.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Finally, we can compute U by the formula σui = Avi orui = 1
σAvi . This gives
U =
(1/√
2 1/√
2
1/√
2 −1/√
2
)So the full SVD is:
A = UΣV T =
(1/√
2 1/√
2
1/√
2 −1/√
2
)(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1289: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1289.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Finally, we can compute U by the formula σui = Avi orui = 1
σAvi . This gives
U =
(1/√
2 1/√
2
1/√
2 −1/√
2
)So the full SVD is:
A = UΣV T =(1/√
2 1/√
2
1/√
2 −1/√
2
)(5 0 00 3 0
) 1/√
2 1/√
2 0
1/√
18 −1/√
19 4/√
182/3 −2/3 −1/3
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1290: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1290.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition UΣV T , then A can berepresented by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k , is obtained by truncating this sum,after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1291: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1291.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition UΣV T , then A can berepresented by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k , is obtained by truncating this sum,after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1292: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1292.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A
has singular value decomposition UΣV T , then A can berepresented by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k , is obtained by truncating this sum,after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1293: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1293.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition
UΣV T , then A can berepresented by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k , is obtained by truncating this sum,after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1294: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1294.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition UΣV T ,
then A can berepresented by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k , is obtained by truncating this sum,after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1295: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1295.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition UΣV T , then A
can berepresented by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k , is obtained by truncating this sum,after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1296: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1296.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition UΣV T , then A can berepresented
by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k , is obtained by truncating this sum,after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1297: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1297.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition UΣV T , then A can berepresented by the
outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k , is obtained by truncating this sum,after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1298: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1298.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition UΣV T , then A can berepresented by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k , is obtained by truncating this sum,after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1299: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1299.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition UΣV T , then A can berepresented by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k , is obtained by truncating this sum,after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1300: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1300.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition UΣV T , then A can berepresented by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix
of rank k , is obtained by truncating this sum,after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1301: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1301.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition UΣV T , then A can berepresented by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k ,
is obtained by truncating this sum,after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1302: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1302.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition UΣV T , then A can berepresented by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k , is obtained by
truncating this sum,after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1303: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1303.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition UΣV T , then A can berepresented by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k , is obtained by truncating this sum,
after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1304: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1304.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition UΣV T , then A can berepresented by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k , is obtained by truncating this sum,after the first
k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1305: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1305.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition UΣV T , then A can berepresented by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k , is obtained by truncating this sum,after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1306: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1306.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
OBS
If A has singular value decomposition UΣV T , then A can berepresented by the outer product expansion
A = σ1u1vT1 + σ2u2vT2 + · · ·+ σnunvTn
The closest matrix of rank k , is obtained by truncating this sum,after the first k terms:
A′ = σ1u1vT1 + σ1u2vT2 + · · ·+ σkukvTk , k < n
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1307: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1307.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.9Let
A =
0 1 1√2 2 0
0 1 1
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
2 2 22 6 22 2 2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1308: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1308.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.9
Let
A =
0 1 1√2 2 0
0 1 1
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
2 2 22 6 22 2 2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1309: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1309.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.9Let
A =
0 1 1√2 2 0
0 1 1
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
2 2 22 6 22 2 2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1310: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1310.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.9Let
A =
0 1 1√2 2 0
0 1 1
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
2 2 22 6 22 2 2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1311: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1311.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.9Let
A =
0 1 1√2 2 0
0 1 1
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
2 2 22 6 22 2 2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1312: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1312.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.9Let
A =
0 1 1√2 2 0
0 1 1
Compute
the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
2 2 22 6 22 2 2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1313: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1313.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.9Let
A =
0 1 1√2 2 0
0 1 1
Compute the singular values and
the singular value decompositionof A.
Solution
The matrix
AAT =
2 2 22 6 22 2 2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1314: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1314.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.9Let
A =
0 1 1√2 2 0
0 1 1
Compute the singular values and the singular value decomposition
of A.
Solution
The matrix
AAT =
2 2 22 6 22 2 2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1315: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1315.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.9Let
A =
0 1 1√2 2 0
0 1 1
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
2 2 22 6 22 2 2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1316: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1316.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.9Let
A =
0 1 1√2 2 0
0 1 1
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
2 2 22 6 22 2 2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1317: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1317.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.9Let
A =
0 1 1√2 2 0
0 1 1
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
2 2 22 6 22 2 2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1318: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1318.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.9Let
A =
0 1 1√2 2 0
0 1 1
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
2 2 22 6 22 2 2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1319: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1319.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Example 12.9Let
A =
0 1 1√2 2 0
0 1 1
Compute the singular values and the singular value decompositionof A.
Solution
The matrix
AAT =
2 2 22 6 22 2 2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1320: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1320.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and λ3 = 0.Consequently, the singular values of A are σ1 = 2
√2, σ2 =
√2 and
σ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1321: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1321.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and λ3 = 0.Consequently, the singular values of A are σ1 = 2
√2, σ2 =
√2 and
σ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1322: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1322.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ =
−λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and λ3 = 0.Consequently, the singular values of A are σ1 = 2
√2, σ2 =
√2 and
σ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1323: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1323.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) =
−λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and λ3 = 0.Consequently, the singular values of A are σ1 = 2
√2, σ2 =
√2 and
σ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1324: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1324.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and λ3 = 0.Consequently, the singular values of A are σ1 = 2
√2, σ2 =
√2 and
σ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1325: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1325.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues
λ1 = 8, λ2 = 2 and λ3 = 0.Consequently, the singular values of A are σ1 = 2
√2, σ2 =
√2 and
σ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1326: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1326.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8,
λ2 = 2 and λ3 = 0.Consequently, the singular values of A are σ1 = 2
√2, σ2 =
√2 and
σ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1327: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1327.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and
λ3 = 0.Consequently, the singular values of A are σ1 = 2
√2, σ2 =
√2 and
σ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1328: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1328.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and λ3 = 0.
Consequently, the singular values of A are σ1 = 2√
2, σ2 =√
2 andσ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1329: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1329.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and λ3 = 0.Consequently,
the singular values of A are σ1 = 2√
2, σ2 =√
2 andσ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1330: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1330.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and λ3 = 0.Consequently, the singular values
of A are σ1 = 2√
2, σ2 =√
2 andσ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1331: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1331.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and λ3 = 0.Consequently, the singular values of A are
σ1 = 2√
2, σ2 =√
2 andσ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1332: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1332.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and λ3 = 0.Consequently, the singular values of A are σ1 = 2
√2,
σ2 =√
2 andσ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1333: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1333.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and λ3 = 0.Consequently, the singular values of A are σ1 = 2
√2, σ2 =
√2 and
σ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1334: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1334.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and λ3 = 0.Consequently, the singular values of A are σ1 = 2
√2, σ2 =
√2 and
σ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1335: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1335.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and λ3 = 0.Consequently, the singular values of A are σ1 = 2
√2, σ2 =
√2 and
σ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1336: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1336.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and λ3 = 0.Consequently, the singular values of A are σ1 = 2
√2, σ2 =
√2 and
σ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1337: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1337.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
The characteristic polynomial is
−λ3 + 10λ2− 16λ = −λ(λ2− 10λ+ 16) = −λ(λ− 8)(λ− 2) = 0
and AAT has eigenvalues λ1 = 8, λ2 = 2 and λ3 = 0.Consequently, the singular values of A are σ1 = 2
√2, σ2 =
√2 and
σ3 = 0
To give the decomposition, we consider the diagonal matrix ofsingular values
Σ =
2√
2 0 0
0√
2 00 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1338: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1338.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Next, we find an orthonormal set of eigenvectors for AAT . Theeigenvalues of AAT are 8, 2, and 0, and since AAT is symmetric weknow that the eigenvectors will be orthogonal. This giveseigenvectors
u1 =
1√62√61√6
; u2 =
−1√3
1√3
− 1√3
u3 =
1√2
0− 1√
2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1339: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1339.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Next, we find an orthonormal set of eigenvectors for AAT . Theeigenvalues
of AAT are 8, 2, and 0, and since AAT is symmetric weknow that the eigenvectors will be orthogonal. This giveseigenvectors
u1 =
1√62√61√6
; u2 =
−1√3
1√3
− 1√3
u3 =
1√2
0− 1√
2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1340: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1340.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Next, we find an orthonormal set of eigenvectors for AAT . Theeigenvalues of AAT
are 8, 2, and 0, and since AAT is symmetric weknow that the eigenvectors will be orthogonal. This giveseigenvectors
u1 =
1√62√61√6
; u2 =
−1√3
1√3
− 1√3
u3 =
1√2
0− 1√
2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1341: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1341.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Next, we find an orthonormal set of eigenvectors for AAT . Theeigenvalues of AAT are 8, 2, and 0, and
since AAT is symmetric weknow that the eigenvectors will be orthogonal. This giveseigenvectors
u1 =
1√62√61√6
; u2 =
−1√3
1√3
− 1√3
u3 =
1√2
0− 1√
2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1342: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1342.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Next, we find an orthonormal set of eigenvectors for AAT . Theeigenvalues of AAT are 8, 2, and 0, and since AAT is symmetric
weknow that the eigenvectors will be orthogonal. This giveseigenvectors
u1 =
1√62√61√6
; u2 =
−1√3
1√3
− 1√3
u3 =
1√2
0− 1√
2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1343: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1343.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Next, we find an orthonormal set of eigenvectors for AAT . Theeigenvalues of AAT are 8, 2, and 0, and since AAT is symmetric weknow that
the eigenvectors will be orthogonal. This giveseigenvectors
u1 =
1√62√61√6
; u2 =
−1√3
1√3
− 1√3
u3 =
1√2
0− 1√
2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1344: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1344.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Next, we find an orthonormal set of eigenvectors for AAT . Theeigenvalues of AAT are 8, 2, and 0, and since AAT is symmetric weknow that the eigenvectors will be orthogonal.
This giveseigenvectors
u1 =
1√62√61√6
; u2 =
−1√3
1√3
− 1√3
u3 =
1√2
0− 1√
2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1345: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1345.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Next, we find an orthonormal set of eigenvectors for AAT . Theeigenvalues of AAT are 8, 2, and 0, and since AAT is symmetric weknow that the eigenvectors will be orthogonal. This giveseigenvectors
u1 =
1√62√61√6
; u2 =
−1√3
1√3
− 1√3
u3 =
1√2
0− 1√
2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1346: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1346.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Next, we find an orthonormal set of eigenvectors for AAT . Theeigenvalues of AAT are 8, 2, and 0, and since AAT is symmetric weknow that the eigenvectors will be orthogonal. This giveseigenvectors
u1 =
1√62√61√6
; u2 =
−1√3
1√3
− 1√3
u3 =
1√2
0− 1√
2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1347: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1347.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Next, we find an orthonormal set of eigenvectors for AAT . Theeigenvalues of AAT are 8, 2, and 0, and since AAT is symmetric weknow that the eigenvectors will be orthogonal. This giveseigenvectors
u1 =
1√62√61√6
;
u2 =
−1√3
1√3
− 1√3
u3 =
1√2
0− 1√
2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1348: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1348.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Next, we find an orthonormal set of eigenvectors for AAT . Theeigenvalues of AAT are 8, 2, and 0, and since AAT is symmetric weknow that the eigenvectors will be orthogonal. This giveseigenvectors
u1 =
1√62√61√6
; u2 =
−1√3
1√3
− 1√3
u3 =
1√2
0− 1√
2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1349: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1349.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Next, we find an orthonormal set of eigenvectors for AAT . Theeigenvalues of AAT are 8, 2, and 0, and since AAT is symmetric weknow that the eigenvectors will be orthogonal. This giveseigenvectors
u1 =
1√62√61√6
; u2 =
−1√3
1√3
− 1√3
u3 =
1√2
0− 1√
2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1350: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1350.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Next, we find an orthonormal set of eigenvectors for AAT . Theeigenvalues of AAT are 8, 2, and 0, and since AAT is symmetric weknow that the eigenvectors will be orthogonal. This giveseigenvectors
u1 =
1√62√61√6
; u2 =
−1√3
1√3
− 1√3
u3 =
1√2
0− 1√
2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1351: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1351.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Next, we find an orthonormal set of eigenvectors for AAT . Theeigenvalues of AAT are 8, 2, and 0, and since AAT is symmetric weknow that the eigenvectors will be orthogonal. This giveseigenvectors
u1 =
1√62√61√6
; u2 =
−1√3
1√3
− 1√3
u3 =
1√2
0− 1√
2
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1352: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1352.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1353: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1353.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1354: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1354.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1355: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1355.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1356: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1356.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now
we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1357: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1357.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find
the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1358: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1358.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors
(the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1359: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1359.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )
byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1360: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1360.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding
an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1361: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1361.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors
of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1362: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1362.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1363: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1363.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1364: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1364.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1365: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1365.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues
of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1366: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1366.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA
are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1367: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1367.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and
since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1368: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1368.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric
we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1369: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1369.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that
the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1370: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1370.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal.
Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1371: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1371.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
U =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
Now we find the right singular vectors (the columns of V )byfinding an orthonormal set of eigenvectors of ATA.
ATA =
2 2√
2 0
2√
2 6 20 2 2
The eigenvalues of ATA are 8, 2, and 0, and since ATA issymmetric we know that the eigenvectors will be orthogonal. Thisgives eigenvectors
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1372: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1372.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
v1 =
1√63√121√12
v2 =
1√3
02√6
v3 =
1√2
−12
12
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1373: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1373.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
v1 =
1√63√121√12
v2 =
1√3
02√6
v3 =
1√2
−12
12
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1374: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1374.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
v1 =
1√63√121√12
v2 =
1√3
02√6
v3 =
1√2
−12
12
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1375: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1375.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
v1 =
1√63√121√12
v2 =
1√3
02√6
v3 =
1√2
−12
12
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1376: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1376.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
v1 =
1√63√121√12
v2 =
1√3
02√6
v3 =
1√2
−12
12
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1377: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1377.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
v1 =
1√63√121√12
v2 =
1√3
02√6
v3 =
1√2
−12
12
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1378: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1378.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
v1 =
1√63√121√12
v2 =
1√3
02√6
v3 =
1√2
−12
12
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1379: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1379.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
V =
1√6
1√3
1√2
3√12
0 −12
1√12− 2√
612
Finally, we can now we verify that we have A = UΣV T
A = UΣV T =1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
2√
2 0 0
0√
2 00 0 0
1√6
1√3
1√2
3√12
0 −12
1√12− 2√
612
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1380: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1380.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
V =
1√6
1√3
1√2
3√12
0 −12
1√12− 2√
612
Finally, we can now we verify that we have A = UΣV T
A = UΣV T =1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
2√
2 0 0
0√
2 00 0 0
1√6
1√3
1√2
3√12
0 −12
1√12− 2√
612
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1381: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1381.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
V =
1√6
1√3
1√2
3√12
0 −12
1√12− 2√
612
Finally, we can now we verify that we have A = UΣV T
A = UΣV T =1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
2√
2 0 0
0√
2 00 0 0
1√6
1√3
1√2
3√12
0 −12
1√12− 2√
612
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1382: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1382.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
V =
1√6
1√3
1√2
3√12
0 −12
1√12− 2√
612
Finally,
we can now we verify that we have A = UΣV T
A = UΣV T =1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
2√
2 0 0
0√
2 00 0 0
1√6
1√3
1√2
3√12
0 −12
1√12− 2√
612
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1383: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1383.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
V =
1√6
1√3
1√2
3√12
0 −12
1√12− 2√
612
Finally, we can now we verify that we have A = UΣV T
A = UΣV T =1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
2√
2 0 0
0√
2 00 0 0
1√6
1√3
1√2
3√12
0 −12
1√12− 2√
612
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1384: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1384.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
V =
1√6
1√3
1√2
3√12
0 −12
1√12− 2√
612
Finally, we can now we verify that we have A = UΣV T
A = UΣV T =
1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
2√
2 0 0
0√
2 00 0 0
1√6
1√3
1√2
3√12
0 −12
1√12− 2√
612
Dr. Marco A Roque Sol Linear Algebra. Session 9
![Page 1385: Linear Algebra. Session 9 - Texas A&M Universityroquesol/Math_304_Fall_2019_Session_9.pdf · Session 9. Abstract Linear Algebra I Singular Value Decomposition (SVD) Complex Eigenvalues](https://reader031.vdocuments.us/reader031/viewer/2022011822/5ec4bf05d175fa4d8b6b27e6/html5/thumbnails/1385.jpg)
Abstract Linear Algebra ISingular Value Decomposition (SVD)
SVD. IntroductionSVD. Examples
SVD. Examples
Put these together to get
V =
1√6
1√3
1√2
3√12
0 −12
1√12− 2√
612
Finally, we can now we verify that we have A = UΣV T
A = UΣV T =1√6− 1√
31√2
2√6
1√3
01√6− 1√
3− 1√
2
2√
2 0 0
0√
2 00 0 0
1√6
1√3
1√2
3√12
0 −12
1√12− 2√
612
Dr. Marco A Roque Sol Linear Algebra. Session 9