Spaces of triangularizable matrices

Abstract

Let \(\mathbb {F}\) be a field. We investigate the greatest possible dimension \(t_n(\mathbb {F})\) for a vector space of n-by-n matrices with entries in \(\mathbb {F}\) and in which every element is triangularizable over the ground field \(\mathbb {F}\). It is obvious that \(t_n(\mathbb {F}) \ge \frac{n(n+1)}{2}\), and we prove that equality holds if and only if \(\mathbb {F}\) is not quadratically closed or \(n=1\), excluding finite fields with characteristic 2. If \(\mathbb {F}\) is infinite and not quadratically closed, we give an explicit description of the solutions with the critical dimension \(t_n(\mathbb {F})\), reducing the problem to the one of deciding for which integers \(k \in \mathopen {[\![}2,n\mathclose {]\!]}\) all k-by-k symmetric matrices over \(\mathbb {F}\) are triangularizable.