Characterizations of k-commutative equalities for some outer generalized inverses

Abstract

In this paper, we present necessary and sufficient conditions for the k-commutative equality $A^k X = X A^k$, where $X$ is an outer generalized inverse of the square matrix A. Also, we give new representations for core EP, DMP, and CMP inverses of square matrices as outer inverses with prescribed null space and range. In addition, we characterize the core EP inverse as the solution of a new system of matrix equations.