Linear spanning sets for matrix spaces

TL;DR

This paper characterizes when the linear span of matrices of the form $A^i S B^j$ covers the entire matrix space, and applies this to finite fields to determine subset cardinalities.

Contribution

It provides necessary and sufficient conditions for the span of $A^i S B^j$ to be the whole matrix space, extending understanding of matrix algebra over fields.

Findings

Derived conditions for spanning the matrix space.

Determined subset cardinalities over finite fields.

Connected algebraic conditions with combinatorial applications.

Abstract

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