Page 463 - Advanced Linear Algebra

Singular Values and the Moore–Penrose Inverse 447

Proof. We leave it to the reader to show that b does indeed satisfy conditions
1)–4) and prove only the uniqueness. Suppose that and satisfy 1)–4) when

substituted for b . Then
~
~² i ³

~ ii
³
~² ii
i iii
~
iii
~² ³

~ ii
~

~

and
~
³
~² i
ii
~
~ ² i ³ i
~ iiii

ii
²
~ ³ i
ii
~
~

~

which shows that ~ .
The MP inverse can also be defined for matrices. In particular, if ( 4 Á ²- , ³
then the matrix operator has an MP inverse ( b . Since this is a linear
(
transformation from - to - , it is just multiplication by a matrix b ~ ) .
(
This matrix is the MP inverse for and is denoted by ( b .
)
(
Since b ~ b and ( ~ ) , the matrix version of Theorem 17.2 implies
( ( ) (
that ( b is completely characterized by the four conditions
b
1) (( ( ~ (
b
b
2) ((( ~ ( b
)
3 (( b is Hermitian
b
)
4 (( is Hermitian
Moreover, if
(~ < ' < i
is the singular-value decomposition of , then
(