Pseudoinverse

I believe that Carl. D. Meyer (p393 ) provides the best definition of the term pseudoinverse. Pseudoinverse is an umbrella term, and there are three types of pseudoinverse:
 * Inner pseudoinverse of $$A$$ is $$X$$ such that $$AXA=A$$
 * Outer pseudoinverse of $$A$$ is $$X$$ such that $$XAX=X$$
 * Reflexivepseudoinverse of $$A$$ is $$X$$ such that $$AXA=X$$ and $$XAX=X$$

It looks like the close term generalized inverse, used by Harville (p107), means inner pseudoinverse.

Important pseudoinverses

 * The quintessential pseudoinverse is the Moore-Penrose Pseudoinverse, which is a reflexive pseudoinverse
 * Another critical pseudoinverse is the Drazin Inverse, which is an outer pseudoinverse.