Multiplicative perturbations of matrices and the generalized triple reverse order law for the Moore-Penrose inverse

For a matrix A, let A(dagger) denote its Moore-Penrose inverse. A matrix M is called a multiplicative perturbation of T is an element of C-mxn if M = ET F* for some E is an element of C-mxm and F is an element of C-nxn. Based on the alternative expression for M as M = (ETT dagger).T.((FTT)-T-dagger)*, the generalized triple reverse order law for the Moore-Penrose inverse is obtained as M-dagger = (((FTT)-T-dagger)*)(dagger).(YY(dagger)TZZ(dagger))(LR-1)(dagger).(ETT dagger)(dagger), where YY(dagger)TZZ(dagger))(LR-1)(dagger), is the weighted Moore-Penrose inverse for certain matrices Y, Z, L and R associated to the triple (T, E, F). Furthermore, it is proved that this weighted Moore-Penrose inverse in the resulting expression for M-dagger can be really replaced with T-dagger if (ETT dagger)(ETT dagger)-E-dagger.T = T.((FTT)-T-dagger)(dagger)((FTT)-T-dagger). In the special case that rank(M) = rank(T) or M is a weak perturbation of T, a simplified version of M-dagger, as well as MM dagger and (MM)-M-dagger, is also derived. (C) 2017 Elsevier Inc. All rights reserved.