New characterizations of the diamond partial order

Baksalary and Hauke introduced the diamond partial order in 1990, which we revisit in this paper. This order was defined on the set of rectangular matrices and is the same as the star and minus partial orders for partial isometries. New ways of describing and studying the diamond partial order are being looked into in this paper. Particularly, we present a new characterization by using an additivity property of the column spaces. Additionally, we also study the relationship between the left (resp., right) star and diamond partial orders. Specifically, we obtain conditions in which the diamond partial order means the left (resp., right) star partial order. The reverse order law for the Moore-Penrose inverse is characterized when $A$ is below $B$ under the diamond partial order. Finally, an interesting way of describing bi-dagger matrices is found.