# pseudoinverse

## Primary tabs

Synonym:
pseudo-inverse, Moore-Penrose pseudoinverse
Type of Math Object:
Definition
Major Section:
Reference

## Mathematics Subject Classification

### Generalized Pseudo-Inverse

As far as I know a matrix B is a (generalized) pseudo-inverse of A if

(1) ABA=A
(2) BAB=B

I am missing (2) in the article.

### Re: Generalized Pseudo-Inverse

There are different notions of pseudo-inverse (or generalized inverse). At the moment I'm using a book called "Generalized Inverses" by Ben-Israel and Greville in which its first chapter you can find all these different notions. They give four conditions,

(1) ABA=A
(2) BAB=B
(3) (AB)*=AB
(4) (BA)*=BA

There's a unique B satisfying the four conditions (a {1,2,3,4}-inverse in Ben-Israel and Greville's notation), which is the Moore-Penrose pseudo-inverse. What TN talks about is a {1,2}-inverse and so on.