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Unformatted text preview: P Gi = Pi Li Qi . (7.9.7) ⎟ ⎠ j It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu Thus (7.9.6) can be written as s ki −1 f (A) = i=1 j =0 D E f (j ) (λi ) (A − λi I)j Gi , j! (7.9.8) and this expression is independent of which similarity is used to reduce A to J. Not only does (7.9.8) prove that f (A) is uniquely defined, but it also provides a generalization of the spectral theorems for diagonalizable matrices given on pp. 517 and 526 because if A is diagonalizable, then each ki = 1 so that (7.9.8) reduces to (7.3.6) on p. 526. Below is a formal summary along with some related properties. T H IG R Spectral Resolution of f (A) For A ∈ C n×n with σ (A) = {λ1 , λ2 , . . . , λs } such that ki = index (λi ), and for a function f : C → C such that f (λi ), f (λi ), . . . , f (ki −1) (λi ) exist for each λi ∈ σ (A) , the value of f (A) is Y P s ki −1 f (A) = i=1 j =0 O C f (j ) (λi ) (A − λi I)j Gi , j! (7.9.9) where the spectral projectors Gi ’s have the following properties. • Gi is the projector onto the generalized eigenspace N (A − λi I)ki along R (A − λi I)ki . • G1 + G 2 + · · · + G s = I . (7.9.10) • Gi Gj = 0 when i = j. (7.9.11) • Ni = (A − λi I)Gi = Gi (A − λi I) is nilpotent of index ki . (7.9.12) • If A is diagonalizable, then (7.9.9) reduces to (7.3.6) on p. 526, and the spectral projectors reduce to those described on p. 517. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 604 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 Proof of (7.9.10)–(7.9.12). Property (7.9.10) results from using (7.9.9) with the function f (z ) = 1, and property (7.9.11) is a consequence of I = P−1 P =⇒ Qi Pj = I 0 if i = j, if i = j. (7.9.13) It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu To prove (7.9.12), establish that (A − λi I)Gi = Gi (A − λi I) by noting that T (7.9.1...
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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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