# 105 ig r max i j aij max i aii j i aij

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

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: explain why k 1 λk−1 k 2 k 1 λk λk−2 · · · λk−1 .. .. .. . . . λk k m−1 λk−m+1 λk−2 k 1 λk−1 D E λk 7.9.6. Determine e A for A = 6 −2 0 2 2 0 8 −2 2 T H . √ 7.9.7. For f (z ) = 4 z − 1, determine f (A) when A = 7.9.8. IG R ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎠ . . . k 2 −3 5 −1 ⎞ −8 11 −2 −9 9 1 . (a) Explain why every nonsingular A ∈ C n×n has a square root. √ (b) Give necessary and suﬃcient conditions for the existence of A when A is singular. Y P 7.9.9. Spectral Mapping Property. Prove that if (λ, x) is an eigenpair for A, then (f (λ), x) is an eigenpair for f (A) whenever f (A) exists. Does it also follow that alg multA (λ) = alg multf (A) (f (λ))? O C 7.9.10. Let f be deﬁned at A, and let λ ∈ σ (A) . Give an example or an explanation of why the following statements are not necessarily true. (a) f (A) is similar to A. (b) geo multA (λ) = geo multf (A) (f (λ)) . (c) indexA (λ) = indexf (A) (f (λ)). 7.9.11. Explain why Af (A) = f (A)A whenever f (A) exists. 7.9.12. Explain why a function f is deﬁned at A ∈ C n×n if and only if f T is deﬁned at AT , and then prove that f (AT ) = f (A) . Why can’t ∗ T ( ) be used in place of ( ) ? Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 614 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 7.9.13. Use the technique of Example 7.9.5 (p. 608) to establish the following identities. (a) eA e−A = I for all A ∈ C n×n . α for all α ∈ C and A ∈ C n×n . (b) eαA = eA iA (c) e = cos A + i sin A for all A ∈ C n×n . It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] 7.9.14. (a) Show that if AB = BA, then eA+B = eA eB . (b) Give an example to show that eA+B = eA eB in general. D E 7.9.15. Find the Hermite interpolation polynomial p(z ) as described in Example 7.9.4 such that p(A) = eA for A = 3 −3 −3 2 −2 −2 1 −1 −1 . T H 7.9.16. The Cayley–Hamilton theorem (pp. 509, 532) says that every A ∈ C n×n satisﬁes its own character...
View Full Document

## This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

Ask a homework question - tutors are online