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Unformatted text preview: + λx3 . . . Axp = xp−1 + λxp O C 2 3 p In other words, the first column x1 in P is a eigenvector for A associated with λ. We already knew there had to be exactly one independent eigenvector for each Jordan block because there are t = dim N (A − λI) Jordan blocks J (λ), but now we know precisely where these eigenvectors are located in P. Vectors x such that x ∈ N (A − λI)g but x ∈ N (A − λI)g−1 are called generalized eigenvectors of order g associated with λ. So P consists of an eigenvector followed by generalized eigenvectors of increasing order. Moreover, the columns of P form a Jordan chain analogous to (7.7.2) on p. 576; i.e., p−i xi = (A − λI) xp implies P must have the form P= p−1 (A − λI) p−2 xp | (A − λI) xp | · · · | (A − λI) xp | xp . (7.8.5) A complete set of Jordan chains associated with a given eigenvalue λ is determined in exactly the same way as Jordan chains for nilpotent matrices are Copyright c 2000 SIAM Buy online from SIAM Buy from 594 Chapter 7 Eigenvalues and Eigenvectors determined except that the nested subspaces Mi defined in (7.7.1) on p. 575 are redefined to be Mi = R (A − λI)i ∩ N (A − λI) for i = 0, 1, . . . , k, (7.8.6) where k = index (λ). Just as in the case of nilpotent matrices, it follows that 0 = Mk ⊆ Mk−1 ⊆ · · · ⊆ M0 = N (A − λI) (see Exercise 7.8.8). Since is a nilpotent linear operator of index k (Example 5.10.4, (A − λI) k / It is illegal to print, duplicate, or distribute this material Please report violations to N ((AλI) ) p. 399), it can be argued that the same process used to build Jordan chains for nilpotent matrices can be used to build Jordan chains for a general eigenvalue λ. Below is a summary of the process adapted to the general case. D E Constructing Jordan Chains For each λ ∈ σ (An×n ) , set Mi = R (A − λI)i ∩ N (A − λI) for i = k − 1, k − 2,...
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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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