P-canonical forms and complete inverses

TL;DR

This paper introduces the complete inverse in associative rings, linking its existence to the Drazin inverse, and shows how to derive the canonical form of the inverse from the matrix's canonical form.

Contribution

It defines the complete inverse, establishes its existence condition via the Drazin inverse, and relates it to the canonical form of matrices.

Findings

Complete inverse exists if and only if Drazin inverse exists.

Complete inverse can be obtained by substituting -k into the canonical form.

Provides a method to compute the canonical form of the complete inverse.

Abstract

This paper describes a new kind of inverse for elements in associative ring, that is the complete inverse, as the unique solution of a certain set of equations. This inverse exists for an element $a$ if and only if the Drazin inverse of $a$ exists. We also show that by plugging in $-k$ for $k$ in the $\mathcal{P}$-canonical form of a square matrix $A$, we get the $\mathcal{P}$-canonical form of the complete inverse of $A$.