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.

Keywords

• weak $m$-WGI
• $W$-weighted $m$-WGI
• minimal rank $W$-weighted weak Drazin inverse

Supporting Institution

Ministry of Science, Technological Development and Innovation, Republic of Serbia

Project Number

451-03-137/2025-03/200124

Ethical Statement

This is original work of authors, the first submission of the paper to journal, and the paper is not submitted for publication elsewhere. The author declares that there is no conflict of interest. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Thanks

The authors are supported by the Ministry of Science, Technological Development and Innovation, Republic of Serbia, grant number 451-03-137/2025-03/200124.

References

1. [1] R. Behera, G. Maharana and J. K. Sahoo, Further results on weighted core-EP inverse of matrices, Results Math. 75, 174, 2020