Solutions of the matrix inequalities $BX{B}^{\ast }<{=}^{-}A$ in the minus partial ordering and $BX{B}^{\ast }<{=}^{L}A$ in the Löwner partial ordering

Abstract

Two matrices $A$ and $B$ of the same size are said to satisfy the minus partial ordering, denoted by $B{⩽}^{-}A$, iff the rank subtractivity equality $\mathrm{rank}(A-B)=\mathrm{rank}(A)-\mathrm{rank}(B)$ holds; two complex Hermitian matrices $A$ and $B$ of the same size are said to satisfy the Löwner partial ordering, denoted by $B{⩽}^{\mathrm{L}}A$, iff the difference $A-B$ is nonnegative definite. In this note, we establish general solution of the inequality $BX{B}^{\ast }{⩽}^{-}A$ induced from the minus partial ordering, and general solution of the inequality $BX{B}^{\ast }{⩽}^{\mathrm{L}}A$ induced from the Löwner partial ordering, respectively, where $(\cdot {)}^{\ast }$ 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 Löwner partial orderings, respectively, and show that these two types of shorted matrices in fact are the same.