On diamond partial order, one-sided star partial orders, and 1MP-inverses

TL;DR

This paper explores new characterizations of the diamond partial order for matrices using inverse properties and decompositions, introduces 1MP-inverses for this order, and investigates one-sided star partial orders with these inverses.

Contribution

It provides novel characterizations of the diamond partial order and extends the use of 1MP-inverses to analyze one-sided star partial orders in matrix theory.

Findings

New characterizations of the diamond partial order using inverse properties.

Use of 1MP-inverses to characterize the diamond partial order.

Analysis of one-sided star partial orders with 1MP- and MP1-inverses.

Abstract

This paper provides some new characterizations of the diamond partial order for rectangular matrices by using properties of inner inverses, minus order, and SVD decompositions. In addition, the recently introduced 1MP generalized inverse and its dual are used to characterize the diamond partial order as a ${\cal G}$-based one. Finally, the one-sided star partial order is investigated by using 1MP- and MP1-inverses.