Numerical Analysis – Numerical Analysis – Eigenvalue and EigenvectorEigenvalue and Eigenvector
Hanyang University
Jong-Il Park
Department of Computer Science and Engineering, Hanyang University
Eigenvalue problemEigenvalue problem
: eigenvalue x : eigenvector ※ spectrum: a set of all eigenvalue
xAx
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Eigenvalue
Eigenvalue
if det x=0(trivial solution) To obtain a non-trivial solution,
det
;Characteristic equation
0)( xIA
0)( IA
0)( IA
0
21
11
22221
11211
nnnn
n
n
aaa
a
aaa
aaa
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Properties of Eigenvalue
1) Trace
2) det
3) If A is symmetric, then the eigenvectors are
orthogonal:
4) Let the eigenvalues of then, the eigenvalues of (A - aI)
n
ii
n
iiiaA
11
i
n
iA
1
jiG
jixx
iij
Ti
,0
nA ,, 21
,,,, 21 aaa n
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Geometrical Interpretation of Eigenvectors
Transformation
: The transformation of an eigenvector
is mapped onto the same line of.
Symmetric matrix orthogonal eigenvectors
Relation to Singular Value
if A is singular 0 {eigenvalues}
xAx Ax
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Eg. Calculating Eigenvectors(I)
Exercise1) ; symmetric, non-singular matrix
( = -1, -6)
2) ; non-symmetric, non-singular matrix
( = -3, -4)
22
25
22
15
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Eg. Calculating Eigenvectors(II)
3) ; symmetric, singular matrix
( = 5, 0)
4) ; non-symmetric, singular matrix
( = 7, 0)
42
21
63
21
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Discussion
symmetric matrix=> orthogonal eigenvectors
singular matrix=> 0 {eigenvalue}
Investigation into SVD
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Similar MatricesSimilar Matrices
Eigenvalues and eigenvectors of similar matrices
Eg. Rotation matrix
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Similarity TransformationSimilarity Transformation
Coordinate transformation x’=Rx, y’=Ry
Similarity transformation y=Ax y’=Ry=RAx=RA(R-1 x’)= RAR-1 x’=Bx’
B = RAR-1
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Numerical Methods(I)
Power method
Iteration formula
for obtaining large
)1()1()1()( kkkk xyAx
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Eg. Power methodEg. Power method
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Numerical Methods (II)
Inverse power method
Iteration formula
for obtaining small
cUyxLc
xAyyxAkk
kkkk
)1()(
)()1()1()(1
,
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Exploiting shifting propertyExploiting shifting property
Let the eigenvalues of then, the eigenvalues of (A - aI)
nA ,, 21
,,,, 21 aaa n
• Finding the maximum eigenvalue with opposite sign after obtaining
• Accelerating the convergence when an approximate eigenvalue is available
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Deflated matricesDeflated matrices
It is possible to obtain eigenvectors one after another Properly assigning the vector x is important Eg. Wielandt’s deflation
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Eg. Using Deflation(I)Eg. Using Deflation(I)
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Eg. Using Deflation(II)Eg. Using Deflation(II)
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Numerical Methods (III)
Hotelling's deflation method
Iteration formula:
given for symmetric matrices deflation from large to small
Tiiiii xxAA 1
ii x,
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Numerical Methods (IV)
Jacobi transformation
Successive diagonalization without changing .
for symmetric matrices
Department of Computer Science and Engineering, Hanyang University
Homework #7 Homework #7
Generate a 9x9 symmetric matrix A by using random number generator(Gaussian distribution with mean=0 and standard deviation=1.1]). Then, compute all eigenvalues and eigenvectors of A using the routines in the book, NR in C. Print the eigenvalues and their corresponding eigenvectors in the descending order. You may use
jacobi(): Obtaining eigenvalues using the Jacobi transformation
eigsrt(): Sorting the results of jacobi()
[Due: Nov. 19]