Drazin inverse

Summary

The Drazin inverse of a square matrix $A$ is a matrix $X$ that commutes with $A$ and satisfies $XAX = X$ and $A^{k+1}X = A^k$ for some $k$. It exists and is unique for every square matrix over a field. It inverts $A$ on the part of the space where $A$ acts invertibly, and it is $0$ on the rest.