On Generalizations of the Reverse Order Law
David T. Barwick, Jimmie D. Gilbert · SIAM Journal on Applied Mathematics · 1974
T. N. E. Greville has shown that the equations $BB^ + A^ * AB = A^ * AB$ and $A^ + ABB^ * A^ * = BB^ * A^ * $ are necessary and sufficient in order that the reverse order law $(AB)^ + = B^ + A^ + $ hold for pseudoinverses of matrices over the field of complex numbers. E. Arghiriade’s results give the more concise formulation that $(AB)^ + = B^ + A^ + $ if and only if $A^ * ABB^ * $ is an $EP_r $ matrix. In this paper, we analyze modifications of the reverse order law which are obtained by replacing one of the pseudo-inverses in the reverse order law by either a normalized generalized inverse or a weak generalized inverse. The results obtained are related to those of Greville.