Special forms of generalized inverses of row block matrices

Abstract

Given a row block matrix [ A, B ], this paper investigates the relations between the generalized
inverse [ A, B ]⁻ and the column block matrix  consisting of two generalized inverses A⁻
and B⁻. The first step of the investigation is to establish a formula for the minimal rank of the
difference [ A, B ]⁻ − , the secondstep is to finda necessary andsufficient condition for
[ A, B ]⁻ =  to holdby letting the minimal rank be zero. Seven types of generalized inverses
of matrices are taken into account.