Generalized core inverses of matrices

TL;DR

This paper introduces two new types of generalized matrix inverses, extending existing concepts like the core inverse and core-EP inverse, with formulas and properties derived using matrix decompositions and powers.

Contribution

The paper presents the $ra{i}{m}$-core inverse and $ra{j}{m}$-core inverse, expanding the theory of matrix inverses beyond previous definitions.

Findings

Defined the $ra{i}{m}$-core inverse and $ra{j}{m}$-core inverse.

Derived formulas and properties using matrix decompositions.

Extended existing inverse concepts to new generalized forms.

Abstract

In this paper, we introduce two new generalized inverses of matrices, namely, the $\bra{i}{m}$-core inverse and the $\pare{j}{m}$-core inverse. The $\bra{i}{m}$-core inverse of a complex matrix extends the notions of the core inverse defined by Baksalary and Trenkler \cite{BT} and the core-EP inverse defined by Manjunatha Prasad and Mohana \cite{MM}. The $\pare{j}{m}$-core inverse of a complex matrix extends the notions of the core inverse and the ${\rm DMP}$-inverse defined by Malik and Thome \cite{MT}. Moreover, the formulae and properties of these two new concepts are investigated by using matrix decompositions and matrix powers.