Spaces of triangularizable matrices
Clément de Seguins Pazzis
TL;DR
The paper investigates the maximal dimension $t_n(\mathbb{F})$ of subspaces of $n\times n$ matrices over a field $\mathbb{F}$ where every matrix is triangularizable over $\mathbb{F}$. It develops an interdisciplinary framework combining adapted vectors, dual-orthogonality, and Atkinson-type theorems to establish when $t_n(\mathbb{F})$ equals the natural bound ${n+1 \choose 2}$, showing this happens precisely when $\mathbb{F}$ is not quadratically closed or when $n=1$, with exceptions for finite characteristic-2 fields. For infinite NRC fields, it gives an explicit description of the $t_n(\mathbb{F})$-dimensional spaces, reducing the problem to determining for which integers $k\in[2,n]$ all $k\times k$ symmetric matrices over $\mathbb{F}$ are triangularizable. In the case of infinite perfect fields of characteristic $2$, it proves $t_n(\mathbb{F})={n+1 \choose 2}$ and classifies irreducible optimal weakly triangularizable subspaces as $\mathop{\mathrm{M}}_1(\mathbb{F})$ and $\mathfrak{sl}_2(\mathbb{F})$, with a decomposition theory for optimal spaces via joins. Collectively, these results illuminate the contrast between real and complex fields, connect to Gerstenhaber-type results, and extend the toolkit for studying spaces of matrices with constrained spectra or triangularizability.
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.