In this paper, we revise the core EP inverse of a square matrix introduced by Prasad and Mohana in [Core EP inverse, Linear and Multilinear Algebra, 62 (3) (2014) 792-802]. Firstly, we give a new representation and a new characterization of the core EP inverse. Then, we study some properties of the core EP inverse by using a representation by block matrices. Secondly, we extend the notion of core EP inverse to rectangular matrices by means of a weighted core EP decomposition. Finally, we study some properties of weighted core EP inverses.

In this paper, we present necessary and sufficient conditions for the k-commutative equality A k X = XA 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.

In this paper, we introduce two new generalized inverses of matrices, namely, the hi; mi-core inverse and the .j; m/-core inverse. The hi; mi-core inverse of a complex matrix extends the notions of the core inverse defined by Baksalary and Trenkler [1] and the core-EP inverse defined by Manjunatha Prasad and Mohana [10]. The .j; m/-core inverse of a complex matrix extends the notions of the core inverse and the DMP-inverse defined by Malik and Thome [9]. Moreover, the formulae and properties of these two new concepts are investigated by using matrix decompositions and matrix powers.