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

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.