Spaces of symmetric matrices containing a nonzero matrix of bounded rank Original Research Article

Let Sn(F) denote the space of all n × n symmetric matrices over the field F. Given a positive integer k such that k < n, let d(n, k, F) be the smallest integer ℓ such that every ℓ dimensional subspace of Sn(F) contains a nonzero matrix whose rank is at most k. It is our purpose to consider d(n, k, F) for real and the field of complex numbers. While the computation of d(n, k, Fthe field of complex numbers) is quite straightforward, we point out the difficulty in evaluating d(n, k, Freal). We obtain partial results regarding d(n, nscript capital lreal), and in particular show that 4less-than-or-equals, slantd(4,2,real).