class_5 - CS205 Class 5 Covered in class: 1, 3, 4, 5....

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CS205 – Class 5 Covered in class: 1, 3, 4, 5. Reading: Heath Chapter 4. Eigenvalues / Eigenvectors Continued 1. The Power Method allows one to compute the largest eigenvalue and eigenvector. Starting from a nonzero vector 0 x , iterate with 1 k k x Ax . a. To see why this works, assume that 0 x is a linear combination of eigenvectors 0 i i i x u where the i u are the eigenvectors of A. Then 2 1 2 0 k k k k x Ax A x A x and so 0 k k k k k i i i i i i i i i i x A x A u A u u   . Now assuming that the largest eigenvalue is 1 , we can write     1 1 1 1 1 1 1 2 2 / k k k k k i i i i i i i i x u u u u   and note that the second term vanishes as k   since 1 / 1 i . Thus as k   , 1 1 1 k k x u   . Moreover     1 1 / k k j j x x for any component j of x. b. If the starting vector 0 i i i x u happens to have 0 i for the largest eigenvalue, the method fails. c. For a real matrix and real 0 x , one can never get complex numbers. d. The largest eigenvalue may be repeated, in which case the final vector may be a linear combination of
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class_5 - CS205 Class 5 Covered in class: 1, 3, 4, 5....

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