**Unformatted text preview: **σ2 , . . . , σr ) contains the
nonzero singular values of A, then
V∗ A∗ AV = D2
0 0
0 , (7.5.9) 2
and this means that (σi , vi ) for i = 1, 2, . . . , r is an eigenpair for A∗ A. In
other words, the nonzero singular values of A are precisely the positive square
roots of the nonzero eigenvalues of A∗ A, and right-hand singular vectors vi of
A are particular eigenvectors of A∗ A. Note that this establishes the uniqueness
of the σi ’s (but not the vi ’s), and pay attention to the fact that the number
of zero singular values of A need not agree with the number of zero eigenvalues
of A∗ A —e.g., A1×2 = (1, 1) has no zero singular values, but A∗ A has one
zero eigenvalue. The same game can be played with AA∗ in place of A∗ A to
argue that the nonzero singular values of A are the positive square roots of Copyright c 2000 SIAM Buy online from SIAM
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554
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] the nonzero eigenvalues of AA∗ , and left-hand singular vectors ui of A are
particular eigenvectors of AA∗ .
Caution! The statement that right-hand singular vectors vi of A are eigenvectors of A∗ A and left-hand singular vectors ui of A are eigenvectors of AA∗
is a one-way street—it doesn’t mean that just any orthonormal sets of eigenvectors for A∗ A and AA∗ can be used as respective right-hand and left-hand
singular vectors for A. The columns vi of any unitary matrix V that diagonalizes A∗ A as in (7.5.9) can serve as right-hand singular vectors for A, but
corresponding left-hand singular vectors ui are constrained by the relationships
Avi = σi ui , i = 1, 2, . . . , r
u∗ A = 0,
i =⇒ i = r + 1, . . . , m =⇒ D
E Avi
Avi
=
, i = 1, 2, . . . , r,
σi
Avi 2
span {ur+1 , ur+2 , . . . , um } = N (A∗ ).
ui = T
H In other words, t...

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