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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.
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
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7.1 Elementary Properties of Eigensystems
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