# T h problem if 1 2 n are the eigenvalues of b explain

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Unformatted text preview: d the diagonal entries of D are the associated eigenvalues. IG R Proof. If A is normal with σ (A) = {λ1 , λ2 , . . . , λk } , then A − λk I is also normal. All normal matrices are RPN (range is perpendicular to nullspace, p. 409), so there is a unitary matrix Uk such that U∗ (A k Y P − λk I)Uk = O C Ck 0 0 0 (by (5.11.15) on p. 408) or, equivalently U∗ AUk = k Ck +λk I 0 0 λk I = Ak−1 0 0 λk I , where Ck is nonsingular and Ak−1 = Ck + λk I. Note that λk ∈ σ (Ak−1 ) (oth/ erwise Ak−1 − λk I = Ck would be singular), so σ (Ak−1 ) = {λ1 , λ2 , . . . , λk−1 } (Exercise 7.1.4). Because Ak−1 is also normal, the same argument can be repeated with Ak−1 and λk−1 in place A and λk to insure the existence of a unitary matrix Uk−1 such that U∗ −1 Ak−1 Uk−1 = k Copyright c 2000 SIAM Ak−2 0 0 λk−1 I , Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 548 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] where Ak−2 is normal and σ (Ak−2 ) = {λ1 , λ2 , . . . , λk−2 } . After k such repetitions, Uk Uk−1 0 · · · U1 0 = U is a unitary matrix such that 0 I 0 I ⎛λ I 0 ··· 0⎞ 1 a1 λ2 Ia2 · · · 0⎟ ⎜0 U∗ AU = ⎜ . . . ⎟ = D, ai = alg multA (λi ) . (7.5.1) .. ⎝. . .⎠ . . . . 0 0 · · · λk Iak Conversely, if there is a unitary matrix U such that U∗ AU = D is diagonal, then A∗ A = UD∗ DU∗ = U = UDD∗ U∗ = AA∗ , so A is normal. D E Caution! While it’s true that normal matrices possess a complete orthonormal set of eigenvectors, not all complete independent sets of eigenvectors of a normal A are orthonormal (or even orthogonal)—see Exercise 7.5.6. Below are some things that are true. T H Properties of Normal Matrices If A is a normal matrix with σ (A) = {λ1 , λ2 , . . . , λk } , then • A is RPN—i.e., R (A) ⊥ N (A) (see p. 408). •...
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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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