Inner-star and star-inner matrices

Abstract

In order to generalize the concepts of the 1MP and MP1 inverses and to solve some systems of matrix equations,
we present the inner-star and star-inner matrices as two new
classes of rectangular matrices. We prove different characterizations and representations of these new matrices.
We obtain the relation between the set of all inner-star (or star-inner) matrices of a given rectangular matrix $A$ and the set of all 1MP (MP1) inverses of $(A^\dag)^*$, and also the relations between
inner-star and star-inner matrices with corresponding nonsingular
matrices. Applications of inner-star and star-inner matrices
in defining two new partial orders and in solving several systems of linear equations are proposed.