Page 169 - Linear Algebra Done Right
P. 169
Polar and Singular-Value Decompositions
0
s 1
M T, (e 1 ,...,e n ), (f 1 ,...,f n ) =
0 . . . s n . 157
In other words, every operator on V has a diagonal matrix with respect
to some orthonormal bases of V, provided that we are permitted to
use two different bases rather than a single basis as customary when
working with operators.
Singular values and the singular-value decomposition have many ap-
plications (some are given in the exercises), including applications in
computational linear algebra. To compute numeric approximations to
the singular values of an operator T, first compute T T and then com-
∗
pute approximations to the eigenvalues of T T (good techniques exist
∗
for approximating eigenvalues of positive operators). The nonnegative
square roots of these (approximate) eigenvalues of T T will be the (ap-
∗
proximate) singular values of T (as can be seen from the proof of 7.28).
In other words, the singular values of T can be approximated without
computing the square root of T T.
∗
