the qr iteration for eigenvalues. ... the intention of the algorithm is to perform a sequence of...
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![Page 1: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/1.jpg)
The QR iteration for eigenvalues
![Page 2: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/2.jpg)
. . .
The intention of the algorithm is to perform a sequence of similarity
transformations on a real matrix so that the limit is a triangular matrix.
![Page 3: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/3.jpg)
. . .
The intention of the algorithm is to perform a sequence of similarity
transformations on a real matrix so that the limit is a triangular matrix.
If this were possible then the eigenvalues would be exactly the
diagonal elements.
![Page 4: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/4.jpg)
But it may not be possible:
![Page 5: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/5.jpg)
But it may not be possible:since • Real matrices may have complex eigenvaluesand• All of the arithmetic in the algorithm is real
![Page 6: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/6.jpg)
But it may not be possible:since • Real matrices may have complex eigenvaluesand• All of the arithmetic in the algorithm is real
There is no way the real numbers can converge to anything other than real numbers.
![Page 7: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/7.jpg)
But it may not be possible:since • Real matrices may have complex eigenvaluesand• All of the arithmetic in the algorithm is real
There is no way the real numbers can converge to anything other than real numbers.
That is: It is impossible for the limit to have numbers with non-zero imaginary parts.
![Page 8: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/8.jpg)
But it may not be possible:since • Real matrices may have complex eigenvaluesand• All of the arithmetic in the algorithm is real
There is no way the real numbers can converge to anything other than real numbers.
That is: It is impossible for the limit to have numbers with non-zero imaginary parts.
If any eigenvalues have non-zero imaginary parts, the sequence will not converge to them.
![Page 9: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/9.jpg)
If any eigenvalues have non-zero imaginary parts, the sequence will not converge to them.
![Page 10: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/10.jpg)
If any eigenvalues have non-zero imaginary parts, the sequence will not converge to them.
Are we dead?
![Page 11: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/11.jpg)
If any eigenvalues have non-zero imaginary parts, the sequence will not converge to them.
Are we dead?
Nope, but we have to modify our expectations.
![Page 12: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/12.jpg)
. . .
Instead of the limit being an upper triangular matrix
![Page 13: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/13.jpg)
. . .
Instead of the limit being an upper triangular matrix
it is block upper triangular
![Page 14: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/14.jpg)
. . .
Instead of the limit being an upper triangular matrix
it is block upper triangular
![Page 15: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/15.jpg)
. . .
Instead of the limit being an upper triangular matrix
it is block upper triangular
The blocks are 2 by 2 and…
![Page 16: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/16.jpg)
. . .
Instead of the limit being an upper triangular matrix
it is block upper triangular
The blocks are 2 by 2 and…the eigenvalues we want are the
complex conjugate pairs of eigenvalues of the blocks
![Page 17: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/17.jpg)
. . .
This actually presents no major troubles.
The blocks are 2 by 2 and…the eigenvalues we want are the
complex conjugate pairs of eigenvalues of the blocks
![Page 18: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/18.jpg)
So this is the algorithm in a mathematical form(as opposed to form representing what happens in
storage):
![Page 19: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/19.jpg)
So this is the algorithm in a mathematical form(as opposed to form representing what happens in
storage):
0. Set A1 = AFor k = 1, 2, …
1. Do a QR factorization of Ak: Ak = QkRk
2. Set Ak+1 = RkQk
![Page 20: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/20.jpg)
This is the algorithm in a programming form:
For k = 1, 2, …1. Do a QR factorization of A: A → QR2. Set A ← RQ
![Page 21: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/21.jpg)
Since Ak = QkRk
QkTAk = Qk
TQkRk= Rk
![Page 22: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/22.jpg)
Since Ak = QkRk
QkTAk = Qk
TQkRk= Rk
but then
Ak+1 = RkQk= QkTAkQk
![Page 23: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/23.jpg)
Since Ak = QkRk
QkTAk = Qk
TQkRk= Rk
but then
Ak+1 = RkQk= QkTAkQk
and since Qk is orthogonal, QkT = Qk
-1 and
![Page 24: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/24.jpg)
Since Ak = QkRk
QkTAk = Qk
TQkRk= Rk
but then
Ak+1 = RkQk= QkTAkQk
and since Qk is orthogonal, QkT = Qk
-1 and
Ak+1 = Qk-1AkQk
![Page 25: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/25.jpg)
Since Ak = QkRk
QkTAk = Qk
TQkRk= Rk
but then
Ak+1 = RkQk= QkTAkQk
and since Qk is orthogonal, QkT = Qk
-1 and
Ak+1 = Qk-1AkQk
Ak+1 is similar to Ak
![Page 26: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/26.jpg)
Ak+1 is similar to Ak
![Page 27: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/27.jpg)
Ak+1 is similar to Ak
is similar to Ak-1
![Page 28: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/28.jpg)
Ak+1 is similar to Ak
is similar to Ak-1
is similar to Ak-2
![Page 29: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/29.jpg)
Ak+1 is similar to Ak
is similar to Ak-1
is similar to Ak-2
. . .
is similar to A1 =A
![Page 30: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/30.jpg)
Ak+1 is similar to Ak
is similar to Ak-1
is similar to Ak-2
. . .
is similar to A1 =A
We have a sequence of similar matricesA1, A2, A3, … tending to a block triangular matrix
whose eigenvalues are easy to obtain.
![Page 31: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/31.jpg)
Not only are the matrices in the sequence similar they are
orthogonally similar - the similarity transformation is orthogonal
![Page 32: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/32.jpg)
Not only are the matrices in the sequence similar they are
orthogonally similar - the similarity transformation is orthogonal
Since orthogonal matrices preserve lengths, this means:
• The matrices of the sequence do not get very large or very small, and
• The computations are done more accurately.
![Page 33: The QR iteration for eigenvalues. ... The intention of the algorithm is to perform a sequence of similarity transformations on a real matrix so that the](https://reader030.vdocuments.us/reader030/viewer/2022032703/56649d015503460f949d3e75/html5/thumbnails/33.jpg)
Let’s see the algorithm in action.
The sizes will be indicated by color.
Since, what will be interesting is seeing the subdiagonal components get smaller, we will use
a logarithmic scale that emphasizes small numbers.
1. (Unshifted) QR2. Corner shifted QR3. Double shift QR