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Let $S$ be a set of $n\times n$ matrices over a field $\mathbb{F}$. We showthat the $\mathbb{F}$-linear span of the words in $S$ of length at most$$2n\log_2n+4n$$ is the full $\mathbb{F}$-algebra generated by $S$. Thisimproves on the $n^2/3+2/3$ bound by Paz (1984) and an $O\left(n^{1.5}\right)$bound of Pappacena (1997).

An improved bound for the length of matrix algebras