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:
  1. Every element of F is non-invertible, and
  2. 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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