A prove that if is real and nonzero then i is not an

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Unformatted text preview: y (A − λI)−1 c = c/(λk − λ). / (b) For an arbitrary vector dn×1 , prove that the eigenvalues of A + cdT agree with those of A except that λk is replaced by λk + dT c. (c) How can d be selected to guarantee that the eigenvalues of A + cdT and A agree except that λk is replaced by a specified number µ? Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu Buy from AMAZON.com 7.1 Elementary Properties of Eigensystems http://www.amazon.com/exec/obidos/ASIN/0898714540 503 7.1.18. Suppose that A is a square matrix. (a) Explain why A and AT have the same eigenvalues. (b) Explain why λ ∈ σ (A) ⇐⇒ λ ∈ σ (A∗ ) . Hint: Recall Exercise 6.1.8. (c) Do these results imply that λ ∈ σ (A) ⇐⇒ λ ∈ σ (A) when A is a square matrix of real numbers? (d) A nonzero row vector y∗ is called a left-hand eigenvector for A whenever there is a scalar µ ∈ C such that y∗ (A − µI) = 0. Explain why µ must be an eigenvalue for A in the “right-hand” sense of the term when A is a square matrix of real numbers. D E 7.1.19. Consider matrices Am×n and Bn×m . (a) Explain why AB and BA have the same characteristic polynomial if m = n. Hint: Recall Exercise 6.2.16. (b) Explain why the characteristic polynomials for AB and BA can’t be the same when m = n, and then explain why σ (AB) and σ (BA) agree, with the possible exception of a zero eigenvalue. T H IG R 7.1.20. If AB = BA, prove that A and B have a common eigenvector. Hint: For λ ∈ σ (A) , let the columns of X be a basis for N (A − λI) so that (A − λI)BX = 0. Explain why there exists a matrix P such that BX = XP, and then consider any eigenpair for P. Y P 7.1.21. For fixed matrices Pm×m and Qn×n , let T be the linear operator on C m×n defined by T(A) = PAQ. (a) Show that if x is a right-hand eigenvector for P and y∗ is a left-hand eigenvector for Q, then xy∗ is an eigenvector for T. (b) Explain why trace (T) = trace (P) trace (Q). O C 7.1.22. Let D = diag (λ1 , λ2 , . . . ,...
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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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