Spaces of triangularizable matrices

TL;DR

This paper determines the maximum dimension of spaces of n-by-n matrices over a field F where every matrix is triangularizable, providing explicit characterizations depending on properties of F.

Contribution

It establishes the exact maximum dimension of such matrix spaces and characterizes when this maximum is achieved based on the field's properties.

Findings

Maximum dimension t_n(F) equals n(n+1)/2 under certain conditions.

Explicit description of matrix spaces achieving the maximum dimension.

Reduction of the problem to triangularizability of symmetric matrices.

Abstract

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