Further properties and representations of the W-weighted m-weak group inverse
Jiale Gao, Qing-Wen Wang, Kezheng Zuo
TL;DR
The paper advances the theory of the $W$-weighted $m$-weak group inverse ($W$-$m$-WG) by establishing new projector relations, delivering multiple representations in terms of established inverses like $A^{D,W}$, $A^{\textcircled{\dag},W}$, and $A^{\textcircled{w},W}$, and offering three concise expressions alongside a singular-value-decomposition–based canonical form. It extends the landscape of generalized inverses by unifying and generalizing the $W$-weighted weak group, $W$-weighted Drazin, and $m$-weak group inverses, and it validates the results with numerical experiments that show high numerical accuracy. The work provides a solid framework for practical computation of $W$-$m$-WG inverses and points toward future analytical directions in perturbation analysis and operator settings, with potential applications in constrained minimization problems. Overall, it broadens both the theoretical and computational toolkit for weighted generalized inverses in complex matrices.
Abstract
The purpose of this paper is to explore more properties and representations of the W-weighted m-weak group (in short, W-m-WG) inverse. We first explore an interesting relation between two projectors with respect to the W-m-WG inverse. Then, the W-m-WG inverse is represented by various generalized inverses including W-weighted Drazin inverse, W-weighted weak group inverse, W-weighted core inverse, etc. We also give three concise explicit expressions for the W-m-WG inverse. Moreover, a canonical form of the W-m-WG inverse is presented in terms of the singular value decomposition. Finally, several numerical examples are designed to illustrate some results given in the paper.