# Chapter7

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Unformatted text preview: AU = T1 and U∗ BU = T2 for some unitary matrix U. Hint: Recall Exercise 7.1.20 (p. 503) along with the development of Schur’s triangularization theorem (p. 508). Y P 7.2.16. For diagonalizable matrices, prove that AB = BA if and only if A and B can be simultaneously diagonalized—i.e., P−1 AP = D1 and P−1 BP = D2 for some P. Hint: If A and B commute, then so do 0 1 P−1 AP = λ0I D and P−1 BP = W X . YZ O C 7.2.17. Explain why the following “proof” of the Cayley–Hamilton theorem is not valid. p(λ) = det (A − λI) =⇒ p(A) = det (A − AI) = det (0) = 0. 7.2.18. Show that the eigenvalues of the ﬁnite diﬀerence matrix (p. 19) ⎛ ⎞ 2 −1 ⎜ −1 2 −1 ⎟ ⎜ ⎟ jπ .. .. .. ⎜ ⎟ are λj = 4 sin2 A=⎜ , 1 ≤ j ≤ n. . . . ⎟ 2(n + 1) ⎝ −1 2 −1 ⎠ −1 2 n×n Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.2 Diagonalization by Similarity Transformations http://www.amazon.com/exec/obidos/ASIN/0898714540 ⎞ ⎛ 0 ⎜ 7.2.19. Let N = ⎜ ⎝ 1 .. . .. . .. . 523 ⎟ ⎟. 1⎠ 0 n×n It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] (a) Show that λ ∈ σ N + NT if and only if iλ ∈ σ N − NT . (b) Explain why N + NT is nonsingular if and only if n is even. (c) Evaluate det N − NT /det N + NT when n is even. 7.2.20. A Toeplitz matrix having the ⎛ c0 ⎜ c1 ⎜ C = ⎜ c2 ⎜. ⎝. . cn−1 form ⎞ c1 c2 ⎟ ⎟ c3 ⎟ .⎟ .⎠ . cn−1 c0 c1 . . . cn−2 cn−1 c0 . . . ··· ··· ··· .. . cn−2 cn−3 · · · c0 D E T H n×n is called a circulant matrix . If p(x) = c0 + c1 x + · · · + cn−1 xn−1 , and if {1, ξ, ξ 2 , . . . , ξ n−1 } are the nth roots of unity, then the results of Exercise 5.8.12 (p. 379) insure that ⎛ p(1) ⎞ 0 ··· 0 ⎜0 Fn CF−1 = ⎜ . n ⎝. . 0 R Y IG p(ξ ) · · · . .. . . . 0 0 . . . ⎟ ⎟ ⎠ · · · p(ξ n−1 ) in which Fn is the Fourier matrix of...
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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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