# Generalized inverse

### From Glossary

Suppose is any matrix. is a
*generalized inverse* of
if is and . Then, a
fundamental theorem for linear equations is:

The equation, , has a solution if and only if for any generalized inverse, , in which case the solutions are of the form:for any

Here is an Example.

The *Moore-Penrose* class additionally requires that
and be symmetric (hermitian, if is
complex). In particular, if ,
is a Moore-Penrose inverse. Note that
, and is a projection matrix for
the subspace ,
since implies
.
Further, projects into its orthogonal complement,
which is the null space of , i.e.
for any .