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Unformatted text preview: n2 A + cos2 A = I for a diagonalizable matrix A. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.3 Functions of Diagonalizable Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 539 It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu 7.3.4. Explain e0 = I for every square zero matrix. 7.3.5. The spectral mapping property for diagonalizable matrices says that if f (A) exists, and if {λ1 , λ2 , . . . , λn } are the eigenvalues of An×n (including multiplicities), then {f (λ1 ), . . . , f (λn )} are the eigenvalues of f (A). (a) Establish this for diagonalizable matrices. ∞ (b) Establish this when an inﬁnite series f (z ) = n=0 cn (z − z0 )n ∞ n deﬁnes f (A) = n=0 cn (A − z0 I) as discussed in (7.3.7). D E 7.3.6. Explain why det eA = etrace(A) . T H 7.3.7. Suppose that for nondiagonalizable matrices Am×m an inﬁnite series ∞ ∞ f (z ) = n=0 cn (z − z0 )n is used to deﬁne f (A) = n=0 cn (A − z0 I)n as suggested in (7.3.7). Neglecting convergence issues, explain why there is a polynomial p(z ) of at most degree m − 1 such that f (A) = p(A). IG R 7.3.8. If f (A) exists for a diagonalizable A, explain why Af (A) = f (A)A. What can you say when A is not diagonalizable? 7.3.9. Explain why eA+B = eA eB whenever AB = BA. Give an example to show that eA+B , eA eB , and eB eA all can diﬀer when AB = BA. Hint: Exercise 7.2.16 can be used for the diagonalizable case. For the general case, consider F(t) = e(A+B)t − eAt eBt and F (t). Y P O C 7.3.10. Show that eA is an orthogonal matrix whenever A is skew symmetric. 7.3.11. A particular electronic device consists of a collection of switching circuits that can be either in an ON state or an OFF state. These electronic switches are allowed to change state at regular time intervals called clock cycles. Suppose that at the end of each clock cycle, 30% of the switches currently in the O...
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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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