A note on the reverse order law for reflexive generalized inverse of multiple matrix products

The reverse order law for reflexive generalized inverses of multiple matrix products was introduced and discussed in [M.Wei, Reverse order laws for generalized inverse of multiple matrix products, Linear Algebra Appl., 293 (1999) 273-288]. There the author derived some necessary and sufficient conditions for the reverse order lawAn{1,2}An-1{1,2}…A1{1,2}=(A1A2…An){1,2},by applying the product singular value decomposition (P-SVD). In this note, we revisited this reverse order law by using the extremal rank relations of generalized Schur complements and a new simpler equivalent condition is obtained in terms of only the ranks of the known matrices.