Solutions of the matrix inequalities $BXB^* <=^- A$ in the minus partial ordering and $BXB^* <=^L A$ in the L\"owner partial ordering

TL;DR

This paper derives general solutions for matrix inequalities involving the minus and L"owner partial orderings, providing explicit formulas for shorted matrices and demonstrating their equivalence.

Contribution

It introduces unified solutions for inequalities in two partial orderings and connects the concepts of shorted matrices under these orderings.

Findings

Explicit solutions for $BXB^* \,\leqslant^{-}A$ and $BXB^* \,\leqslant^{L}A$.

Closed-form expressions for shorted matrices relative to the range of B.

Demonstration that the shorted matrices in both orderings are identical.

Abstract

Two matrices $A$ and $B$ of the same size are said to satisfy the minus partial ordering, denoted by $B\leqslant^{-}A$, iff the rank subtractivity equality ${\rm rank}(\, A - B\,) = {\rm rank}(A) -{\rm rank}(B)$ holds; two complex Hermitian matrices $A$ and $B$ of the same size are said to satisfy the L\"owner partial ordering, denoted by $B\leqslant^{\rm L} A$, iff the difference $A - B$ is nonnegative definite. In this note, we establish general solution of the inequality $BXB^{*} \leqslant^{-}A$ induced from the minus partial ordering, and general solution of the inequality $BXB^{*} \leqslant^{\rm L} A$ induced from the L\"owner partial ordering, respectively, where $(\cdot)^{*}$ denotes the conjugate transpose of a complex matrix. As consequences, we give closed-form expressions for the shorted matrices of $A$ relative to the range of $B$ in the minus and L\"owner partial orderings, respectively, and show that these two types of shorted matrices in fact are the same.