This talk concerns linear spaces F of nxn-matrices over a field K of
characteristic zero, with the following two properties:
-
Every element of F is non-invertible, 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 Kn
of dimensions d and d-1, respectively,
then the space of all matrices mapping U into V is a maximal singular
space---called a `compression space' for obvious reasons---of dimension
d(d-1)+n(n-d).
-
If n is odd, then the space of all skew-symmetric nxn-matrices is
maximal singular; it has dimension n(n-1)/2.
The dimensions of these maximal singular spaces are quadratic in
n. An ingeneous construction of n-dimensional 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.
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