#### Vol. 13, No. 6, 2019

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An improved bound for the lengths of matrix algebras

### Yaroslav Shitov

Vol. 13 (2019), No. 6, 1501–1507
DOI: 10.2140/ant.2019.13.1501
##### Abstract

Let $S$ be a set of $n×n$ 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).

##### Keywords
matrix theory, finite-dimensional algebras, generating sets
Primary: 15A03
Secondary: 15A30