Composite outer inverses for rectangular matrices

Abstract

Various compositions of the Drazin inverse, the group inverse or the core-EP inverse with the Moore-Penrose inverse have investigated  last years. Solving some type of matrix equations, we introduce three new generalized inverses of a rectangular matrix, which are called the OMP, MPO and MPOMP inverses, because the outer inverse and the Moore-Penrose inverse are incorporated in their denition. As a
consequence, the notion of DMP, MPD, CMP and MPCEP inverses for a square matrix are covered by one general denition and extended to a rectangular matrix. We propose a common term, composite outer inverses, to denote such compositions of outer inverses and the Moore-Penrose inverse. Characterizations of the OMP, MPO and MPOMP inverses are derived as well as some properties of projectors determined by these new inverses. We establish maximal classes of matrices for which the representations of composite outer inverses are valid. Also, the integral and limit representations for OMP, MPO and MPOMP inverses are investigated. Possible applications of composite outer inverses are given too and interesting topics for further research are considered.

Key words: Outer inverse, Moore-Penrose inverse, DMP inverse, CMP inverse, integral representation, limit representation.