On Characterizations of W-weighted DMP and MPD Inverses

TL;DR

This paper explores advanced matrix inverses, specifically W-weighted DMP and MPD inverses, providing new characterizations, generalizations, and applications in solving matrix equations and establishing order laws.

Contribution

It introduces a general class of solutions for matrix equations, generalizes the W-weighted Drazin inverse, and derives new properties and expressions for W-weighted DMP and MPD inverses.

Findings

Derived equivalent properties for minimal rank W-weighted weak Drazin inverse.

Established projection-based characterizations and new expressions for MPD and DMP inverses.

Applied results to reverse and forward order laws in matrix equations.

Abstract

Recently, the weak Drazin inverse and its characterization have been crucial studies for matrices of index k. In this article, we have revisited W-weighted DMP and MPD inverses and constructed a general class of unique solutions to certain matrix equations. Moreover, we have generalized the W-weighted Drazin inverse of Meng, 2017 using the minimal rank Wweighted weak Drazin inverse. In addition to that, we have derived several equivalent properties of W-weighted DMP and MPD inverses for minimal rank W-weighted weak Drazin inverse of rectangular matrices. Furthermore, some projection-based results are discussed for the characterization of minimal rank W-weighted Drazin inverse, along with some new expressions that are derived for MPD and DMP inverses. Thereby, we have elaborated certain expressions of the perturbation formula for W-weighted weak MPD and DMP inverses. As an application, we establish the reverse and forward order laws using the W-weighted weak Drazin inverse and the minimal rank W-weighted weak Drazin inverse, and apply these results to solve certain matrix equation.