Generalized inverse

Summary

A generalized inverse of a matrix $A$ is a matrix $G$ with $AGA = A$. Every matrix over a field has one, even when it is not square or not invertible. Generalized inverses give solutions of consistent linear systems $Ax = b$ in the form $x = Gb$. The Moore–Penrose inverse is a generalized inverse with three more conditions that make it unique. The Drazin inverse is a related kind of inverse for square matrices, though it does not satisfy $AGA = A$ in general.