About the group inverse and Moore-Penrose inverse of a product (Q2640665)

Let A be a matrix over a ring. Suppose that \(P,Q,P',Q'\) are matrices over the ring such that \(P'PA=A=AQQ'.\) It is shown that if A has a group inverse \(A^{\#}\), then PAQ has a group inverse iff \(AA^{\#}QPA+1- AA^{\#}\) is invertible. An analogous result for Moore-Penrose inverses is also given. These results are then applied to several familiar factorizations of matrices such as the Schur, polar, and singular value decomposition.