We study the Steinberg variety associated to matrix Schubert varieties, and develop a Robinson-Schensted type correspondence, τ↔(Λ,Q,P). Here τ is a partial permutation of size p×q, Λ
Nearby in the stack
an admissible signed Young diagram of size
p+q
, and
P
(resp.
Q
) a standard Young tableau of size
p
(resp.
q
) whose shape is determined by
Λ
. 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
(Λ,Q,P)↦(Λ∨,P,Q)
corresponds to projective duality on matrix Schubert varieties.