Involutions and fixed-point-free involutions arise naturally as representatives for certain Borel orbits in invertible matrices. Similarly, partial involutions and partial fixed-point-free involutions represent certain Borel orbits in matrices which are not necessarily invertible. Inclusion relations among Borel orbit closures induce a partial order on these discrete parameterizing sets. In this dissertation we investigate the associated order complex of these posets. In particular, we prove that the order complex of the Bruhat poset of Borel orbit closures is shellable in symmetric as well as skew-symmetric matrices.