  We consider the varieties of singular $m \times m$ complex matrices which may
be either general, symmetric or skew-symmetric (with $m$ even). For these
varieties we have shown in another paper that they had compact "model
submanifolds", for the homotopy types of the Milnor fibers which are classical
symmetric spaces in the sense of Cartan. In this paper we use these models,
combined with results due to a number of authors concerning the Schubert
decomposition of Lie groups, symmetric spaces via the Cartan model together
with Iwasawa decomposition to give a cell decomposition of the global Milnor
fibers. The Schubert decomposition is in terms of "unique ordered
factorizations" of matrices in the Milnor fibers as products of "
pseudo-rotations". In the case of symmetric or skew-symmetric matrices, this
factorization has the form of iterated "Cartan conjugacies" by
pseudo-rotations. The "Schubert cycles" (the closures of the Schubert cells),
are images of products of suspensions of projective spaces (complex, real, or
quaternionic). For general or skew-symmetric matrices the Schubert cycles have
fundamental classes, and for symmetric matrices $\mod 2$ classes, giving a
basis for the homology, corresponding to the cohomology generators for the
symmetric spaces. For general matrices the duals of the Schubert cycles are
given as explicit monomials in the generators of the exterior cohomology
algebra; and for symmetric matrices they are related to Stiefel-Whitney classes
of an associated vector bundle. Furthermore, for any matrix singularity of
these types the pull-backs of these cohomolgy classes generate a characteristic
subalgebra of the cohomology of the Milnor fiber. These results extend to
exceptional orbit hypersurfaces, complements and links.
