Estimation of Eigenvalues Largest eigenvalue Power Method Smallest

Estimation of Eigenvalues ü Largest eigenvalue: Power Method ü Smallest eigenvalue: Inverse Power Method ü All Eigenvalues: ü Inverse power method with shift ü Faddeev-Leverrier method ü QR Decomposition

Computation of Eigenvalues

QR-Decomposition a(1) a(2) a(j) q(1) q(2) a(n) q(j) q(n) r(1) r(2) r(j) r(n)

QR-Decomposition: Proof by Induction a(1) q(1) r(1)

QR-Decomposition: Proof by Induction 0 q(1) q(2) a(2) r(2)

QR-Decomposition: proof by induction

QR-Decomposition: proof by induction ü Proceeding this way up to step n, all n columns of Q and all the elements of R can be computed. This concludes the proof that, A = QR can be constructed! ü However, the algorithm in proof is tedious and inefficient! ü It is easier to construct the Q and R independently, directly from A using Gram-Schmidt orthogonalization!

QR decomposition by Gram-Schmidt Orthogonalization 8

Gram-Schmidt Orthogonalization 9

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Gram-Schmidt Orthogonalization 13

QR-Decomposition: Calculating Eigenvalues

QR-Decomposition: Algorithm

Example: QR-Decomposition A=

Example: QR-Decomposition

Example: QR-Decomposition

Example: Eigenvalues by Similarity Transformation k 0 1 2 3 4 5 6 7 Ak = Rk. Qk 3. 0000 2. 0000 7. 9545 0. 6331 0. 0792 8. 1420 0. 0640 0. 0016 8. 1556 0. 0055 0. 0000 8. 1568 0. 0004 0. 0000 8. 1568 0. 0000 8. 1569 0. 0000 4. 0000 5. 0000 2. 3043 0. 9420 -0. 1323 1. 7215 0. 6990 -0. 0328 1. 6505 0. 6656 -0. 0086 1. 6414 0. 6587 -0. 0024 1. 6398 0. 6570 -0. 0007 1. 6395 0. 6565 -0. 0002 1. 6394 0. 6564 -0. 0001 Qk 1. 0000 -0. 0792 0. 2941 0. 1034 0. 2166 0. 4214 0. 1589 0. 2983 0. 4442 0. 1788 0. 3199 0. 4502 0. 1845 0. 3259 0. 4519 0. 1861 0. 3276 0. 4524 0. 1866 0. 3281 0. 4525 0. 1867 0. 6396 0. 4264 0. 9968 0. 0793 0. 0099 1. 0000 0. 0079 0. 0002 1. 0000 0. 0007 0. 0000 1. 0000 0. 0001 0. 0000 1. 0000 0. 0000 -0. 1980 0. 6730 -0. 7126 -0. 0757 0. 9765 -0. 2018 -0. 0078 0. 9988 -0. 0482 -0. 0007 0. 9999 -0. 0130 -0. 0001 1. 0000 -0. 0036 0. 0000 1. 0000 -0. 0010 0. 0000 1. 0000 -0. 0003 Rk -0. 7428 0. 3714 0. 5571 -0. 0257 0. 2004 0. 9794 -0. 0006 0. 0482 0. 9988 0. 0000 0. 0130 0. 9999 0. 0000 0. 0036 1. 0000 0. 0010 1. 0000 0. 0003 1. 0000 4. 6904 0. 0000 7. 9801 0. 0000 8. 1423 0. 0000 8. 1557 0. 0000 8. 1568 0. 0000 8. 1569 0. 0000 6. 6092 1. 1481 0. 0000 2. 3703 0. 7721 0. 0000 1. 7269 0. 6863 0. 0000 1. 6509 0. 6645 0. 0000 1. 6415 0. 6587 0. 0000 1. 6398 0. 6570 0. 0000 1. 6395 0. 6565 0. 0000 e (%) 1. 7056 -0. 2375 0. 1857 -0. 0546 0. 2723 0. 1623 0. 2199 0. 4116 0. 1790 0. 2986 0. 4416 0. 1845 0. 3199 0. 4495 0. 1861 0. 3259 0. 4517 0. 1866 0. 3276 0. 4523 0. 1867 866 34. 92 11. 08 3. 1 0. 88 0. 25 0. 07