The $m$-weak core inverse

TL;DR

This paper introduces the $m$-weak core inverse, a new generalized inverse for matrices of arbitrary index, unifying several existing inverses and exploring its properties through matrix decompositions.

Contribution

It proposes the $m$-weak core inverse, extending the core inverse and related inverses, and analyzes its properties and characterizations.

Findings

Defines the $m$-weak core inverse for matrices of arbitrary index.

Shows that it generalizes the core-EP and WC inverses.

Provides properties and characterizations using matrix decompositions.

Abstract

Since the day the core inverse has been known in a paper of Bakasarly and Trenkler, it has been widely researched. So far, there are four generalizations of this inverse for the case of matrices of an arbitrary index, namely, the BT inverse, the DMP inverse, the core-EP inverse and the WC inverse. In this paper we introduce a new type of generalized inverse for a matrix of arbitrary index to be called $m$-weak core inverse which generalizes the core-EP inverse, the WC inverse, and therefore the core inverse. We study several properties and characterizations of the $m$-weak core inverse by using matrix decompositions.