Small maximal spaces of non-invertible matrices
Jan Draisma (Basel)
This talk concerns linear spaces F of nxn-matrices over a field K of
characteristic zero, with the following two properties:
F is then called a `maximal singular space'.
It is fairly easy to construct maximal singular spaces:
Every element of F is non-invertible, and
- F is not properly contained in a bigger space satisfying 1.
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
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
If n is odd, then the space of all skew-symmetric nxn-matrices is
maximal singular; it has dimension n(n-1)/2.
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.
back to TU/e Combinatorial Theory Seminar