Large affine spaces of non-singular matrices

Abstract:

Let $\mathbb {K}$ be an arbitrary (commutative) field with at least three elements. It was recently proven that an affine subspace of $\operatorname {M}_n(\mathbb {K})$ consisting only of non-singular matrices must have a dimension less than or equal to $\binom {n}{2}$. Here, we classify, up to equivalence, the subspaces whose dimension equals $\binom {n}{2}$. This is done by classifying, up to similarity, all the $\binom {n}{2}$-dimensional linear subspaces of $\operatorname {M}_n(\mathbb {K})$ consisting of matrices with no non-zero invariant vector, reinforcing a classical theorem of Gerstenhaber. Both classifications only involve the quadratic structure of the field $\mathbb {K}$.