  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.
