  We study the Steinberg variety associated to matrix Schubert varieties, and
develop a Robinson-Schensted type correspondence,
$\tau\leftrightarrow(\Lambda,\mathsf Q,\mathsf P)$. Here $\tau$ is a partial
permutation of size $p\times q$, $\Lambda$ an admissible signed Young diagram
of size $p+q$, and $\mathsf P$ (resp. $\mathsf Q$) a standard Young tableau of
size $p$ (resp. $q$) whose shape is determined by $\Lambda$. By embedding the
matrix Schubert variety into a Schubert variety, we find a close relationship
between the combinatorics of the classical Robinson-Schensted-Knuth
correspondence and our bijection. We also show that an involution
$(\Lambda,\mathsf Q,\mathsf P)\mapsto(\Lambda^\vee,\mathsf P,\mathsf Q)$
corresponds to projective duality on matrix Schubert varieties.
