A weak $m$-WGI for rectangular matrices

Abstract

Inspired by significant results regarding the $m$-weak group inverse ($m$-WGI) and its generalization to rectangular matrices, known as the $W$-weighted $m$-WGI ($W$-$m$-WGI), we investigate the solvability of a new and more general matrix equations that extend those previously used to introduce the weak $m$-WGI and $W$-$m$-WGI.As a solution of new system, we define the $W$-weighted weak $m$-WGI ($W$-$w$-$m$-WGI) in terms of a minimal rank $W$-weighted weak Drazin inverse (min-r $W$-WDI).Since the weak $m$-WGI, the $W$-$m$-WGI, and the $W$-weighted Drazin inverse ($W$-DI) arise as particular cases of the proposed $W$-w-$m$-WGI, this framework establishes a broader and more unified class of generalized inverses.Different properties, expressions and characterizations of the $W$-w-$m$-WGI are presented.As application of the $W$-w-$m$-WGI, we derive solution formulas for several classes of linear equations, demonstrating that the proposed inverse provides a unified and effective tool for treating singular and weighted linear systems beyond the scope of existing generalized inverses.