New classes of more general weighted outer inverses

Abstract

According to known definition in the literature, the W-weighted outer inverses of A is defined as the outer inverses of WAW. We introduce and study more general weighted outer inverses of rectangular complex matrices. Introduced families of outer inverses include previous research in two aspects from two approaches. In the first approach, for given complex matrices A, M, N, B, C, we define the -weighted -inverse of A and prove that it is just the outer inverse of MAN with the range and null space . Thus, the -weighted -inverse generalizes the notion of the weighted outer inverse and, finally, the notion of the outer inverse with known image and kernel. We give equivalent conditions for the existence and representations of the -weighted -inverse. Integral and limit representations of weighted generalized inverses can be derived as corollaries. Inspired by corresponding representations of the W-weighted Drazin inverse and the weighted core-EP inverse, in the second approach, we investigate the expressions , and . Particularly, conditions under which these expressions become W-weighted outer inverses of A are analysed.