Lengths of families of matrices allowing a minimal polynomial of given degree

Abstract

Let $S$ be a family generating $\mathrm{Mat}_n(\mathbb{F})$ as an algebra over a field $\mathbb{F}$. If $\mathrm{dim} \mathbb{F}[A]=\delta > 1$d with some $A \in S$, then the products of length at most \[ (2n - \delta) \cdot \lfloor n/\delta \rfloor - (\lfloor n/\delta \rfloor^2 -1) \cdot (\delta +1) + n - 2 \cdot \lfloor (2 \delta- 1)/n \rfloor,  \]

with multipliers in $S$, contain the full algebra $\mathrm{Mat}_n(\mathbb{F})$  in their $\mathbb{F}$-linear span.