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

For a matrix A, let A† denote its Moore-Penrose inverse. A matrix M is called a multiplicative perturbation of T∈Cm×n if M=ETF⁎ for some E∈Cm×m and F∈Cn×n. Based on the alternative expression for M as M=(ETT†)⋅T⋅(FT†T)⁎, the generalized triple reverse order law for the Moore-Penrose inverse is obtained asM†=((FT†T)⁎)†⋅(YY†TZZ†)LR−1†⋅(ETT†)†, where (YY†TZZ†)LR−1† 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† can be really replaced with T† if(ETT†)†ETT†⋅T=T⋅(FT†T)†(FT†T). In the special case that rank(M)=rank(T) or M is a weak perturbation of T, a simplified version of M†, as well as MM† and M†M, is also derived.