Theoretical analysis demonstrates generalized weighted weak group inverses for rectangular matrices, highlighting a unified algebraic framework for solving complex singular linear systems.
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.
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Mosić et al. (2026) studied this question.
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