Chapter7

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Unformatted text preview: herwise λ1 is another eigenvalue with the same magnitude as λ1 . Consider f (z ) = (z/λ1 )n , and use the spectral representation (7.3.6) along with |λi /λ1 | < 1 for i = 2, 3, . . . , k to conclude that A λ1 Y P n = f (A) = f (λ1 )G1 + f (λ2 )G2 + · · · + f (λk )Gk = G1 + O C λ2 λ1 n G2 + · · · + λk λ1 (7.3.16) n Gk → G1 as n → ∞. Consequently, (An x0 /λn ) → G1 x0 ∈ N (A − λ1 I) for all x0 . So if 1 G1 x0 = 0 or, equivalently, x0 ∈ R (A − λ1 I), then An x0 /λn converges to an / 1 eigenvector associated with λ1 . This means that the direction of An x0 tends toward the direction of an eigenvector because λn acts only as a scaling factor 1 to keep the length of An x0 under control. Rather than using λn , we can scale 1 An x0 with something more convenient. For example, An x0 (for any vector norm) is a reasonable scaling factor, but there are even better choices. For vectors v, let m(v) denote the component of maximal magnitude, and if there is more 74 While the development of the power method was considered to be a great achievement when R. von Mises introduced it in 1929, later algorithms relegated its computational role to that of a special purpose technique. Nevertheless, it’s still an important idea because, in some way or another, most practical algorithms for eigencomputations implicitly rely on the mathematical essence of the power method. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 534 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 than one maximal component, let m(v) be the first maximal component—e.g., m(1, 3, −2) = 3, and m(−3, 3, −2) = −3. It’s clear that m(αv) = αm(v) for all scalars α. Suppose m(An x0 /λn ) → γ. Since (An /λn ) → G1 , we see that 1 1 An x0 (An /λn )x0 G1 x0 1 = lim = =x n→∞ m(An x0 ) n→∞ m(An x0 /λn ) γ 1 lim It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] is an eigen...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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