Page 462 - Advanced Linear Algebra

446 Advanced Linear Algebra

We state without proof the following uniqueness facts and refer the reader to
[48] for details. If  and if the eigenvalues are distinct, then 7 is
uniquely determined up to multiplication on the right by a diagonal matrix of the
form + ~ diag ²' ÁÃÁ' ³ with ' ~ . If  , then is never uniquely
8
((

determined. If ~ ~ , then for any given there is a unique . Thus, we
8
7
see that, in general, the singular-value decomposition is not unique.
The Moore–Penrose Generalized Inverse
Singular values lead to a generalization of the inverse of an operator that applies
to all linear transformations. The setup is the same as in Figure 17.1. Referring
b
to that figure, we are prompted to define a linear transformation ¢= ¦ < by
 for
b "
#~ F

€ for
since then
² b ³ O º" ÁÃÁ" » ~

² b ³ O ~ »
º" b ÁÃÁ"
and
² b ³ O ~ ÁÃÁ# »
º#
² b ³ O ~ »
º# b ÁÃÁ#
Hence, if ~ ~ , then b ~ c . The transformation b is called the
Moore–Penrose generalized inverse or Moore–Penrose pseudoinverse of .

We abbreviate this as MP inverse.
b
Note that the composition is the identity on the largest possible subspace of
< on which any composition of the form could be the identity, namely, the
orthogonal complement of the kernel of . A similar statement holds for the

composition b . Hence, b is as “close” to an inverse for as is possible.

We have said that if is invertible, then b ~ c . More is true: If is

b
injective, then ~ and so b is a left inverse for . Also, if is surjective,

then b is a right inverse for . Hence the MP inverse b generalizes the one-

sided inverses as well.
Here is a characterization of the MP inverse.
Theorem 17.2 Let B ²<Á = ³ . The MP inverse b of is completely

characterized by the following four properties:
b
1) ~
b
b
2) ~ b
)
3 b is Hermitian
)
b
4 is Hermitian