RANK CONDITIONS FOR GENERALIZED INVERSES OF PARTITIONED MATRICES
George Marsaglia, George P. H. Styan · 2016
SUMMARY. The usual line of attack on the problem of generalized inverses of partitioned matrices goes like this : One takas the Schur formula that applies when everything is nonsingular, substitutes generalized inverses for regular inverses, then multiplies out to see what happens. The result is a set of c 3nditions which roughly say : If the formula works then it works. We suggest a diffarent approach that gives necessary and sufficient conditions for the Schur formula to hold for generalized, reflexive and Moore-Penrose g-inverses. The conditions aro simple, involve rank, and apply to matrices over any field (except for certain fields where there is no Moore-Penrose g-inverse).