An improved bound for the lengths of matrix algebras

Abstract

Let $S$ be a set of $n\times n$ matrices over a field $\mathbb{F}$. We show that the $\mathbb{F}$-linear span of the words in $S$ of length at most

$$2n{log}_{2}n+4n$$

is the full $\mathbb{F}$-algebra generated by $S$. This improves on the $\frac{n^2}{3}+\frac{2}{3}$ bound by Paz (1984) and an $O\left(n^{3∕2}\right)$ bound of Pappacena (1997).