211 sequentially extend sk1 with sets sk2 sk3

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Unformatted text preview: I)k+1 . • N (A − λI)k = N (A − λI)k+1 . • R (A − λI)k ∩ N (A − λI)k = 0. • T H rank (A − λI)k = rank (A − λI)k+1 . C n = R (A − λI)k ⊕ N (A − λI)k . IG R Y P It is understood that index (µ) = 0 if and only if µ ∈ σ (A) . The Jordan form for A ∈ C n×n is derived by digesting the distinct eigenvalues in σ (A) = {λ1 , λ2 , . . . , λs } one at a time with a core-nilpotent decomposition as follows. If index (λ1 ) = k1 , then there is a nonsingular matrix X1 such that L1 0 , (7.8.1) X−1 (A − λ1 I)X1 = 1 0 C1 O C where L1 is nilpotent of index k1 and C1 is nonsingular (it doesn’t matter whether C1 or L1 is listed first, so, for the sake of convenience, the nilpotent block is listed first). We know from the results on nilpotent matrices (p. 579) that there is a nonsingular matrix Y1 such that ⎞ ⎛ N (λ ) 0 ··· 0 1 ⎜ − Y1 1 L1 Y1 = N(λ1 ) = ⎜ ⎝ 0 . . . 0 1 N2 (λ1 ) · · · . .. . . . 0 0 . . . ⎟ ⎟ ⎠ · · · Nt1 (λ1 ) is a block-diagonal matrix that is characterized by the following features. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 588 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 ⎞ ⎛ 0 ⎜ Every block in N(λ1 ) has the form N (λ1 ) = ⎜ ⎝ 1 .. . .. . .. . 1 0 ⎟ ⎟. ⎠ It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu There are t1 = dim N (L1 ) = dim N (A − λ1 I) such blocks in N(λ1 ). The number of i × i blocks of the form N (λ1 ) contained in N(λ1 ) is νi (λ1 ) = rank Li−1 − 2 rank Li + rank Li+1 . But C1 in (7.8.1) is 1 1 1 p nonsingular, so rank (Lp ) = rank ((A − λ1 I) ) − rank (C1 ), and thus the 1 number of i × i blocks N (λ1 ) contained in N(λ1 ) can be expressed as νi (λ1 ) = ri−1 (λ1 ) − 2ri (λ1 ) + ri+1 (λ1 ), Now, Q1 = X1 Y1 0 0I equivalently, D E ri (λ1 ) = rank (A − λ1 I)i . λ 0 is nonsingular, and Q−1 (A − λ1 I)Q1 = N(0 1 ) C 1 1 N(...
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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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