Large affine spaces of non-singular matrices (Q2838076)

The purpose of this paper is to classify, up to equivalence, the subspaces whose dimension is \(\binom{n}{2}\), by classifying, up to similarity, all the \(\binom{n}{2}\)-dimensional linear subspaces of the algebra \(\text{M}_n(\mathbb K)\) (\(n \times n\) matrices with entries in an arbitrary (commutative) field \(\mathbb K\)) consisting of matrices with no non-zero invariant vector. This strengthens a classical theorem of \textit{M. Gerstenhaber} [Am. J. Math. 80, 614--622 (1958; Zbl 0085.26204)], both classifications only involving the quadratic structure of \(\mathbb K\).