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Two matrices $A$ and $B$ of the same size are said to satisfy the minuspartial ordering, denoted by $B\leqslant^{-}A$, iff the rank subtractivityequality ${\rm rank}(\, A - B\,) = {\rm rank}(A) -{\rm rank}(B)$ holds; twocomplex Hermitian matrices $A$ and $B$ of the same size are said to satisfy theL\"owner partial ordering, denoted by $B\leqslant^{\rm L} A$, iff thedifference $A - B$ is nonnegative definite. In this note, we establish generalsolution of the inequality $BXB^{*} \leqslant^{-}A$ induced from the minuspartial 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. Asconsequences, 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 thesame.

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