  Matrix Schubert varieties are certain varieties in the affine space of square
matrices which are determined by specifying rank conditions on submatrices. We
study these varieties for generic matrices, symmetric matrices, and upper
triangular matrices in view of two applications to algebraic statistics: we
observe that special conditional independence models for Gaussian random
variables are intersections of matrix Schubert varieties in the symmetric case.
Consequently, we obtain a combinatorial primary decomposition algorithm for
some conditional independence ideals. We also characterize the vanishing ideals
of Gaussian graphical models for generalized Markov chains.
  In the course of this investigation, we are led to consider three related
stratifications, which come from the Schubert stratification of a flag variety.
We provide some combinatorial results, including describing the stratifications
using the language of rank arrays and enumerating the strata in each case.
