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

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 meyer@ncsu.edu 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). •...
View Full Document

Ask a homework question - tutors are online