Pseudo Principal Pivot Transform: The Group Inverse Case

TL;DR

This paper derives formulas for the group inverse of block matrices and extends the pseudo principal pivot transform concept, replacing the Moore-Penrose inverse with the group inverse, aiming to facilitate future research in matrix analysis.

Contribution

It provides new formulas for the group inverse of block matrices and extends the pseudo principal pivot transform to this context.

Findings

Formulas for the group inverse of block matrices are established.

Extension of the pseudo principal pivot transform using group inverses.

Potential applications in advanced matrix analysis and inverse problems.

Abstract

In this short note, we prove a formula for the group inverse of a block matrix and consider the pseudo principal pivot transform expressed in terms of group inverses. Extensions of the usual principal pivot transform, where the usual inverse is replaced by the Moore-Penrose inverse, were considered in the literature. The objective here is to derive the corresponding formulae for the group generalized inverse. These are expected to be useful in our future work.