Some new results on core partial order and strong core orthogonal matrices

Abstract

Recently, Ferreyra and Malik (Some new results on the core partial order, Linear and Multilinear Algebra, DOI: 10.1080/03081087.2020.1841078) give an example to show that $A{\overset{\tiny\textcircled{\#}}\leq}B$ does not imply $(B-A)^{\tiny\textcircled{\#}}=B^{\tiny\textcircled{\#}}-A^{\tiny\textcircled{\#}}$, and put forward  an open question: Let $A,B {\in}{\mathbb{C}}^{\!^{\rm GM}}_n$, can $A{\overset-\leq}B$ and $(B-A)^{\tiny\textcircled{\#}}=B^{\tiny\textcircled{\#}}-A^{\tiny\textcircled{\#}}$ ${\Rightarrow}$ $A{\overset{\tiny\textcircled{\#}}\leq}B$ be true? In this paper, the above problem will be completely solved. We also give some necessary and sufficient conditions for core partial order, and we give some new characterizations of strong core orthogonality.