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5.12 Singular Value Decomposition 413

T
This means that U A(S 2 )isan ellipsoid whose k th semiaxis has length
σ k . Because orthogonal transformations are isometries (length preserving trans-
formations), U T can only affect the orientation of A(S 2 ), so A(S 2 )is also an
ellipsoid whose k th semiaxis has length σ k . Furthermore, (5.12.3) implies that
T
the ellipsoid U A(S 2 )isin standard position—i.e., its axes are directed along
T
the standard basis vectors e k . Since U maps U A(S 2 )to A(S 2 ), and since
Ue k = U ∗k , it follows that the axes of A(S 2 ) are directed along the left-hand
singular vectors defined by the columns of U. Therefore, the k th semiaxis of
A(S 2 )is σ k U ∗k . Finally, since AV = UD implies AV ∗k = σ k U ∗k , the right-
hand singular vector V ∗k is a pointon S 2 that is mapped to the k th semiaxis
3
vector on the ellipsoid A(S 2 ). The picture in looks like Figure 5.12.1.
σ 2 U ∗2

1
V ∗2
V ∗1

σ 1 U ∗1
V ∗3

A
σ 3 U ∗3
Figure 5.12.1
The degree of distortion of the unit sphere under transformation by A
is therefore measured by κ 2 = σ 1 /σ n , the ratio of the largest singular value
to the smallest singular value. Moreover, from the discussion of induced ma-
trix norms (p. 280) and the unitary invariance of the 2-norm (Exercise 5.6.9),

max Ax = A = UDV T
= D = σ 1

x 2 =1 2 2 2 2
and
1 1 1
min Ax = = = = σ n .
2
T
x 2 =1 A −1 VD −1 U D −1
2 2 2
In other words, longest and shortest vectors on A(S 2 )have respective lengths

σ 1 = A and σ n =1/ A −1
(this justifies Figure 5.2.1 on p. 281), so
2
2

κ 2 = A
A −1
. This is called the 2-norm condition number of A. Differ-
2 2
ent norms result in condition numbers with different values but with more or
less the same order of magnitude as κ 2 (see Exercise 5.12.3), so the qualitative
information about distortion is the same. Below is a summary.