b for an arbitrary vector dn1 prove that the

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Unformatted text preview: A is nonsingular, and if (λ, x) is an eigenpair for A, show that λ−1 , x is an eigenpair for A−1 . (b) For all α ∈ σ (A), prove that x is an eigenvector of A if and / only if x is an eigenvector of (A − αI)−1 . 7.1.10. Show that if (λ, x) is an eigenpair for A, then (λk , x) is an eigenpair for Ak for each positive integer k. (b) If p(x) = α0 + α1 x + α2 x2 + · · · + αk xk is any polynomial, then we define p(A) to be the matrix (a) p(A) = α0 I + α1 A + α2 A2 + · · · + αk Ak . Show that if (λ, x) is an eigenpair for A, then (p(λ), x) is an eigenpair for p(A). Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 502 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 7.1.11. Explain why (7.1.14) ⎞ Gerschgorin’s theorem on p. 498 implies that in ⎛ 1 0 −2 0 0 5 0 0 4 A = ⎝ 0 12 −0 −0 ⎠ must have at least two real eigenvalues. Cor1 0 1 It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] roborate this fact by computing the eigenvalues of A. 7.1.12. If A is nilpotent ( Ak = 0 for some k ), explain why trace (A) = 0. Hint: What is σ (A)? D E 7.1.13. If x1 , x2 , . . . , xk are eigenvectors of A associated with the same eigenvalue λ, explain why every nonzero linear combination T H v = α1 x1 + α2 x2 + · · · + αn xn is also an eigenvector for A associated with the eigenvalue λ. IG R 7.1.14. Explain why an eigenvector for a square matrix A cannot be associated with two distinct eigenvalues for A. Y P 7.1.15. Suppose σ (An×n ) = σ (Bn×n ) . Does this guarantee that A and B have the same characteristic polynomial? 7.1.16. Construct 2 × 2 examples to prove the following statements. O C (a) λ ∈ σ (A) and µ ∈ σ (B) =⇒ λ + µ ∈ σ (A + B) . (b) λ ∈ σ (A) and µ ∈ σ (B) =⇒ λµ ∈ σ (AB) . 7.1.17. Suppose that {λ1 , λ2 , . . . , λn } are the eigenvalues for An×n , and let (λk , c) be a particular eigenpair. (a) For λ ∈ σ (A) , explain wh...
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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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