Further properties and representations of the W-weighted m-weak group inverse

TL;DR

This paper investigates the properties, representations, and explicit formulas of the W-weighted m-weak group inverse, connecting it with other generalized inverses and providing practical examples.

Contribution

It introduces new properties and representations of the W-weighted m-weak group inverse, including explicit formulas and a canonical form using singular value decomposition.

Findings

Relation between projectors and the W-m-WG inverse

Representation of W-m-WG inverse via generalized inverses

Explicit expressions and canonical form of the inverse

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.