An improved bound for the length of matrix algebras

TL;DR

This paper establishes a tighter upper bound on the length of matrix words needed to generate the entire matrix algebra, improving previous bounds significantly.

Contribution

It provides an improved upper bound of approximately 2n log₂ n + 4n for the length of matrix words generating the full algebra, surpassing earlier bounds.

Findings

New bound of 2n log₂ n + 4n for matrix algebra generation

Improvement over Paz's and Pappacena's bounds

Full algebra generated within shorter matrix words

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 $$2n\log_2n+4n$$ is the full $\mathbb{F}$-algebra generated by $S$. This improves on the $n^2/3+2/3$ bound by Paz (1984) and an $O\left(n^{1.5}\right)$ bound of Pappacena (1997).