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# 111 7113 use krylovs method to determine the

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Unformatted text preview: rting with any nonzero vector bn×1 , let v1 (x) = xk − j =0 αj xj be the minimum polynomial for b relative to A, and let K1 = b | Ab | · · · | Ak−1 b n×k be the associated Krylov matrix. Notice that rank (K1 ) = k (by deﬁnition of the minimum polynomial for b ). If C1 is the k × k companion matrix of v (x) as described in (7.11.6), then direct multiplication shows that Y P O C K1 C1 = AK1 . (7.11.7) If k = n, then K−1 AK1 = C1 , so v1 (x) must be the characteristic polynomial 1 for A, and there is nothing more to do. If k < n, then use any n × (n − k ) matrix K1 such that K2 = K1 | K1 n×n is nonsingular, and use (7.11.7) to write AK2 = AK1 | AK1 = K1 | K1 Therefore, K−1 AK2 = 2 C1 0 X A2 C1 0 X A2 , where X A2 = K−1 AK1 . 2 , and hence c(x) = det (xI − A) = det (xI − C1 )det (xI − A2 ) = v1 (x) det (xI − A2 ). Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 650 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 Repeat the process on A2 . If the Krylov matrix on the second time around is nonsingular, then c(x) = v1 (x)v2 (x); otherwise c(x) = v1 (x)v2 (x) det (xI − A3 ) for some matrix A3 . Continuing in this manner until a nonsingular Krylov matrix is obtained—say at the mth step—produces a nonsingular matrix K such that ⎞ ⎛ It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] K −1 AK= ⎝ ··· .. . C1 .⎠ . . Cm = H, (7.11.8) where the Cj ’s are companion matrices, and thus c(x) = v1 (x)v2 (x) · · · vm (x). D E Note: All companion matrices are upper-Hessenberg matrices as described in Example 5.7.4 (p. 350)—e.g.,...
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