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Unformatted text preview: 3) implies P−1 Gi = 0 · · · Qi · · · 0 and Gi P = 0 · · · Pi · · · 0 . Use this with (7.9.4) to observe that (A − λi I)Gi = Pi Li Qi = Gi (A − λi I). Now Nj = (Pi Li Qi )j = Pi Lj Qi i i for j = 1, 2, 3, . . . , D E and thus Ni is nilpotent of index ki because Li is nilpotent of index ki . Example 7.9.1 Example 7.9.2 A coordinate-free version of the representation in (7.9.3) results by separating the first-order terms in (7.9.9) from the higher-order terms to write ⎡ ⎤ ki −1 (j ) s f (λi ) j ⎦ ⎣f (λi )Gi + f (A) = Ni . j! i=1 j =1 T H IG R Using the identity function f (z ) = z produces a coordinate-free version of the Jordan decomposition of A in the form s A= λ i G i + Ni , i=1 and this is the extension of (7.2.7) on p. 517 to the nondiagonalizable case. Another version of (7.9.9) results from lumping things into one matrix to write s Y P ki −1 f (j ) (λi )Zij , f (A) = where Zij = i=1 j =0 (A − λi I)j Gi . j! O C (7.9.14) The Zij ’s are often called the component matrices or the constituent matrices. Problem: Describe f (A) for functions f defined at A = 6 −2 0 2 2 0 8 −2 2 . Solution: A is block triangular, so it’s easy to see that λ1 = 2 and λ2 = 4 are the two distinct eigenvalues with index (λ1 ) = 1 and index (λ2 ) = 2. Thus f (A) exists for all functions such that f (2), f (4), and f (4) exist, in which case f (A) = f (2)G1 + f (4)G2 + f (4)(A − 4I)G2 . The spectral projectors could be computed directly, but things are easier if some judicious choices of f are made. For example, f (z ) = 1 ⇒ I = f (A) = G1 + G2 f (z ) = (z − 4)2 ⇒ (A − 4I)2 = f (A) = 4G1 Copyright c 2000 SIAM =⇒ G1 = (A − 4I)2 /4, G2 = I − G1 . Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.9 Functions of Nondiagonalizable Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 605 It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu Now...
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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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