Maximally transitive semigroups of $n\times n$ matrices

TL;DR

This paper proves that for both real and complex matrices, there exist pairs of matrices whose generated semigroup is dense in the space of all n×n matrices, revealing a fundamental property of matrix semigroups.

Contribution

The paper establishes the existence of pairs of matrices generating dense subsemigroups in both real and complex cases, advancing understanding of matrix semigroup structure.

Findings

Existence of pairs generating dense subsemigroups in real matrices

Existence of pairs generating dense subsemigroups in complex matrices

Density results hold for all n×n matrices

Abstract

We prove that, in both real and complex cases, there exists a pair of matrices that generates a dense subsemigroup of the set of $n\times n$ matrices.