Page 459 - Advanced Linear Algebra
P. 459
Chapter 17
Singular Values and the Moore–Penrose
Inverse
Singular Values
Let and be finite-dimensional inner product spaces over or and let
d
s
=
<
B ²<Á = ³. The spectral theorem applied to i can be of considerable help
i
in understanding the relationship between and its adjoint . This relationship
is shown in Figure 17.1. Note that and can be decomposed into direct sums
<
=
<~ ( l ) and = ~ * l +
in such a manner that and ¢( ¦ * i ¢* ¦ ( act symmetrically in the sense
that
and ¢" ª # i ¢# ª "
i
Also, both and are zero on and , respectively.
+
)
i
We begin by noting that ²<³ is a positive Hermitian operator. Hence, if
B
~ rk ² ³ ~ rk ² i ³, then has an ordered orthonormal basis
<
8 ~ ²" Á ÃÁ" Á" b ÁÃÁ" ³
i
of eigenvectors for , where the corresponding eigenvalues can be arranged
so that
Ä b ~ ~Ä~
i
² ³
The set ²" b ÁÃÁ" ³ is an ordered orthonormal basis for ker ² ³ ~ ker
and so ²" ÁÃÁ" ³ is an ordered orthonormal basis for ker ² ³ ~ im ² i . ³
