Writing ax x as a i x 0 shows that the vectors of

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Unformatted text preview: of (7.1.1). What we really need are scalars λ and nonzero vectors x that satisfy Ax = λx. Writing Ax = λx as (A − λI) x = 0 shows that the vectors of interest are the nonzero vectors in N (A − λI) . But N (A − λI) contains nonzero vectors if and only if A − λI is singular. Therefore, the scalars of interest are precisely the values of λ that make A − λI singular or, equivalently, the λ ’s for which det (A − λI) = 0. 66 These observations motivate the definition of eigenvalues and eigenvectors. D E Eigenvalues and Eigenvectors T H For an n × n matrix A, scalars λ and vectors xn×1 = 0 satisfying Ax = λx are called eigenvalues and eigenvectors of A, respectively, and any such pair, (λ, x), is called an eigenpair for A. The set of distinct eigenvalues, denoted by σ (A) , is called the spectrum of A. • λ ∈ σ (A) ⇐⇒ A − λI is singular ⇐⇒ det (A − λI) = 0. • x = 0 x ∈ N (A − λI) is the set of all eigenvectors associated with λ. From now on, N (A − λI) is called an eigenspace for A. • Nonzero row vectors y∗ such that y∗ (A − λI) = 0 are called lefthand eigenvectors for A (see Exercise 7.1.18 on p. 503). IG R (7.1.3) Y P Geometrically, Ax = λx says that under transformation by A, eigenvectors experience only changes in magnitude or sign—the orientation of Ax in n is the same as that of x. The eigenvalue λ is simply the amount of “stretch” or “shrink” to which the eigenvector x is subjected when transformed by A. Figure 7.1.1 depicts the situation in 2 . O C Ax = λx x Figure 7.1.1 66 The words eigenvalue and eigenvector are derived from the German word eigen , which means owned by or peculiar to. Eigenvalues and eigenvectors are sometimes called characteristic values and characteristic vectors, proper values and proper vectors, or latent values and latent vectors. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.1 Elementary Properties of Eigensystems http://www.amazon.com/exec/obidos/ASIN/08...
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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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