A note on spaces of symmetric matrices (Q996309)

A vector subspace \(X\subseteq M_n\) is said to be of constant rank \(r\), provided that all its nonzero members have rank \(r\). A classical problem here is to determine how large such a subspace can be. The present authors continue this kind of study. They confine themselves to real symmetric matrices, and obtain a sharp upper and lower bound on the maximal dimension of vector subspaces of constant rank \((n-1)\) and of constant rank \((n-2)\). Upper and lower bound differ only by \(1\). Moreover, conditions are supplied, which force optimality of lower, respectively, of upper bound. Similar estimates are also obtained for complex Hermitian matrices.