This talk concerns linear spaces F of nxnmatrices over a field K of
characteristic zero, with the following two properties:

Every element of F is noninvertible, and
 F is not properly contained in a bigger space satisfying 1.
F is then called a `maximal singular space'.
It is fairly easy to construct maximal singular spaces:

If U and V are subspaces of K^{n}
of dimensions d and d1, respectively,
then the space of all matrices mapping U into V is a maximal singular
spacecalled a `compression space' for obvious reasonsof dimension
d(d1)+n(nd).

If n is odd, then the space of all skewsymmetric nxnmatrices is
maximal singular; it has dimension n(n1)/2.
The dimensions of these maximal singular spaces are quadratic in
n. An ingeneous construction of ndimensional maximal singular spaces
appeared in the literature a few decades ago, and lead Fillmore, Laurie,
and Radjavi to put forward the question of whether there exist even
smaller ones.
I present a linear sufficient condition for a given singular space to
be maximal, and show how to use this condition to prove maximality of
certain singular spaces of fixed dimension 8, for infinitely many n.
