T h example 753 interlaced eigenvalues for a hermitian

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Unformatted text preview: IG R Eigenvectors corresponding to distinct eigenvalues are orthogonal. In other words, N (A − λi I) ⊥ N (A − λj I) • Y P for λi = λj . (7.5.2) The spectral theorems (7.2.7) and (7.3.6) on pp. 517 and 526 hold, but the spectral projectors Gi on p. 529 specialize to become orthogonal projectors because R (A − λi I) ⊥ N (A − λi I) for each λi . O C Proof of (7.5.2). If A is normal, so is A − λj I, and hence A − λj I is RPN. ∗ Consequently, N (A − λj I) = N (A − λj I) —recall (5.11.14) from p. 408. If (λi , xi ) and (λj , xj ) are distinct eigenpairs, then (A − λj I)∗ xj = 0, and 0 = x∗ (A − λj I)xi = x∗ Axi − x∗ λj xi = (λi − λj )x∗ xi implies 0 = x∗ xi . j j j j j Several common types of matrices are normal. For example, real-symmetric and hermitian matrices are normal, real skew-symmetric and skew-hermitian matrices are normal, and orthogonal and unitary matrices are normal. By virtue of being normal, these kinds of matrices inherit all of the above properties, but it’s worth looking a bit closer at the real-symmetric and hermitian matrices because they have some special eigenvalue properties. If A is real symmetric or hermitian, and if (λ, x) is an eigenpair for A, then x∗ x = 0, and λx = Ax implies λx∗ = x∗ A∗ , so x∗ x(λ − λ) = x∗ (λ − λ)x = x∗ Ax − x∗ A∗ x = 0 =⇒ λ = λ. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.5 Normal Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 549 It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] In other words, eigenvalues of real-symmetric and hermitian matrices are real. A similar argument (Exercise 7.5.4) shows that the eigenvalues of a real skewsymmetric or skew-hermitian matrix are pure imaginary numbers. Eigenvectors for a hermitian A ∈ C n×n may have to contain complex numbers, but a real-symmetric matrix possesses a complete orthonormal set of real eigenvectors (all quantities...
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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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