Linear spanning sets for matrix spaces

Abstract

Necessary and sufficient conditions are given on matrices $\mathit{A}$, $\mathit{B}$ and $\mathit{S}$, having entries in some field $\mathbb{F}$ and suitable dimensions, such that the linear span of the terms $\mathit{A}^i \mathit{SB}^j$ over $\mathbb{F}$ is equal to the whole matrix space. This result is then used to determine the cardinality of subsets of $\mathbb{F}[\mathit{A}]\mathit{S} \mathbb{F}[\mathit{B}]$ when $\mathbb{F}$ is a $\mathbf{finite}$ field.