This naive approach fails because x1 r a i means a ix2

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 ) + λ1 I 0 Q−1 AQ1 = 1 where 0 C1 + λ1 I = T H J(λ1 ) 0 0 A1 . or, (7.8.2) The upper-left-hand segment J(λ1 ) = N(λ1 ) + λ1 I has the block-diagonal form ⎛ J (λ ) 0 11 J2 (λ1 ) ⎜0 J(λ1 ) = ⎜ . . ⎝. . . . 0 0 IG R ··· ··· .. . 0 0 . . . ⎞ ⎟ ⎟ ⎠ with J (λ1 ) = N (λ1 ) + λ1 I. · · · Jt1 (λ1 ) Y P The matrix J(λ1 ) is called the Jordan segment associated with the eigenvalue λ1 , and the individual blocks J (λ1 ) contained in J(λ1 ) are called Jordan blocks associated with the eigenvalue λ1 . The structure of the Jordan segment J(λ1 ) is inherited from Jordan structure of the associated nilpotent matrix L1 . ⎞ ⎛λ 1 O C 1 ⎜ Each Jordan block looks like J (λ1 ) = N (λ1 ) + λ1 I = ⎜ ⎝ .. . .. . .. . 1 λ1 ⎟ ⎟. ⎠ There are t1 = dim N (A − λ1 I) such Jordan blocks in the segment J(λ1 ). The number of i × i Jordan blocks J (λ1 ) contained in J(λ1 ) is νi (λ1 ) = ri−1 (λ1 ) − 2ri (λ1 ) + ri+1 (λ1 ), where ri (λ1 ) = rank (A − λ1 I)i . Since the distinct eigenvalues of A are σ (A) = {λ1 , λ2 , . . . , λs } , the distinct eigenvalues of A − λ1 I are σ (A − λ1 I) = {0, (λ2 − λ1 ), (λ3 − λ1 ), . . . , (λs − λ1 )}. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.8 Jordan Form http://www.amazon.com/exec/obidos/ASIN/0898714540 589 Couple this with the fact that the only eigenvalue for the nilpotent matrix L1 in (7.8.1) is zero to conclude that σ (C1 ) = {(λ2 − λ1 ), (λ3 − λ1 ), . . . , (λs − λ1 )}. Therefore, the spectrum of A1 = C1 +λ1 I in (7.8.2) is σ (A1 ) = {λ2 , λ3 , . . . , λs }. This means that the core-nilpotent decomposition process described above can be repeated on A1 − λ2 I to produce a nonsingular matrix Q2 such that It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu Q−1 A1 Q2 = 2 J(λ2 ) 0 0 A2 , where σ (A2 ) = {λ3 , λ4 , ....
View Full Document

Ask a homework question - tutors are online