10 11 7 8 8 5 y p solution each matrix has exactly

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Unformatted text preview: al task. Fortunately, there are some theoretical tools to help determine how many linearly independent eigenvectors a given matrix possesses. O C Independent Eigenvectors Let {λ1 , λ2 , . . . , λk } be a set of distinct eigenvalues for A. • If {(λ1 , x1 ), (λ2 , x2 ), . . . , (λk , xk )} is a set of eigenpairs for A, then S = {x1 , x2 , . . . , xk } is a linearly independent set. (7.2.3) • If Bi is a basis for N (A − λi I), then B = B1 ∪B2 ∪· · ·∪Bk is a linearly independent set. (7.2.4) Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 512 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 Proof of (7.2.3). Suppose S is a dependent set. If the vectors in S are arranged so that M = {x1 , x2 , . . . , xr } is a maximal linearly independent subset, then r xr+1 = αi xi , i=1 It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] and multiplication on the left by A − λr+1 I produces r r αi (Axi − λr+1 xi ) = 0= i=1 αi (λi − λr+1 ) xi . D E i=1 Because M is linearly independent, αi (λi − λr+1 ) = 0 for each i. Consequently, αi = 0 for each i (because the eigenvalues are distinct), and hence xr+1 = 0. But this is impossible because eigenvectors are nonzero. Therefore, the supposition that S is a dependent set must be false. T H Proof of (7.2.4). The result of Exercise 5.9.14 guarantees that B is linearly independent if and only if IG R Mj = N (A − λj I) ∩ N (A − λ1 I) + N (A − λ2 I) + · · · + N (A − λj −1 I) = 0 for each j = 2, 3, . . . , k. Suppose we have 0 = x ∈ Mj for some j. Then Ax = λj x and x = v1 + v2 + · · · + vj −1 for vi ∈ N (A − λi I), which implies Y P j −1 j −1 (λi − λj )vi = i=1 j −1 λ i vi − λ j i=1 vi = Ax − λj x = 0. i=1 O C By (7.2.3), the vi ’s are linearly independent, and hence λi − λj = 0 for each i = 1, 2, . . . , j ...
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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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