The Forward Order Law for Weighted Moore–Penrose Inverse of Multiple Matrix Products

The forward order law for generalized inverses has potential applications in numerical linear algebra, control theory, and statistics. This topic has attracted considerable attention. In this article, we present a necessary and sufficient condition for the forward order law (A1A2⋯An)Mn+1,M1†=(A1)M2,M1†(A2)M3,M2†⋯(An)Mn+1,Mn† by using the rank equalities of some generalized Schur complements. The topic is relevant and timely, and the results generalize existing work on unweighted and two-product cases.