Consequently 1 2 1 1 1 1 1 1 t 2 xy g1 t

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Unformatted text preview: P−1 = I translates to T P−1 P = I =⇒ Yi Xj = I when i = j, 0 when i = j, =⇒ k i=1 Gi = I, and G2 i = Gi , Gi G j = 0 when i = j. To establish that R (Gi ) = N (A − λi I), use R (AB) ⊆ R (A) (Exercise 4.2.12) T and Yi Xi = I to write T T R (Gi ) = R(Xi Yi ) ⊆ R (Xi ) = R(Xi Yi Xi ) = R(Gi Xi ) ⊆ R (Gi ). Thus R (Gi ) = R (Xi ) = N (A − λi I). To show N (Gi ) = R (A − λi I), use k A = j =1 λj Gj with the already established properties of the Gi ’s to conclude ⎛ ⎞ Gi (A − λi I) = Gi ⎝ k j =1 Copyright c 2000 SIAM k λ j Gj − λ i Gj ⎠ = 0 =⇒ R (A − λi I) ⊆ N (Gi ). j =1 Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 518 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 But we already know that N (A − λi I) = R (Gi ), so It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] dim R (A − λi I) = n − dim N (A − λi I) = n − dim R (Gi ) = dim N (Gi ), and therefore, by (4.4.6), R (A − λi I) = N (Gi ). Conversely, if there exist matrices Gi satisfying (7.2.8)–(7.2.10), then A must be diagonalizable. To see this, note that (7.2.8) insures dim R (Gi ) = dim N (A − λi I) = geo multA (λi ) , k k while (7.2.9) implies R (Gi ) ∩ R (Gj ) = 0 and R i=1 Gi = i=1 R (Gi ) (Exercise 5.9.17). Use these with (7.2.10) in the formula for the dimension of a sum (4.4.19) to write D E n = dim R (I) = dim R (G1 + G2 + · · · + Gk ) = dim [R (G1 ) + R (G2 ) + · · · + R (Gk )] = dim R (G1 ) + dim R (G2 ) + · · · + dim R (Gk ) T H = geo multA (λ1 ) + geo multA (λ2 ) + · · · + geo multA (λk ) . k Since geo multA (λi ) ≤ alg multA (λi ) and i=1 alg multA (λi ) = n, the above equation insures that geo multA (λi ) = alg multA (λi ) for each i, and, by (7.2.5), this means A is diagonalizable. IG R Simple Eigenvalues and Projectors If x and y∗ are respective right-hand and left-hand eigen...
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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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