New Representations of the Group Inverse of 2×2 Block Matrices

Abstract

This paper presents a full rank factorization of a $2\times 2$ block matrix without any restriction concerning the group inverse. Applying this factorization, we obtain an explicit representation of the group inverse in terms of four individual blocks of the partitioned matrix without certain restriction. We also derive some important coincidence theorems, including the expressions of the group inverse with Banachiewicz-Schur forms.