Generating Sets of Stochastic Matrices

TL;DR

This paper explores the concept of generating sets for stochastic matrices, providing conditions for divisibility, constructing such sets in low dimensions, and establishing bounds on the number of factors needed for generation.

Contribution

It formalizes divisibility in stochastic matrices, offers necessary and sufficient conditions in low dimensions, and constructs generating sets with bounds on factors required.

Findings

All 2x2 stochastic matrices are divisible.

A sufficient and necessary condition for divisibility in 3x3 matrices.

Upper bounds established for the number of factors in generating sets.

Abstract

This paper introduces the concept of a generating set for stochastic matrices -- a subset of matrices whose repeated composition generates the entire set. Understanding such generating sets requires specifying the "indivisible elements" and "building blocks" within the set, which serve as fundamental components of the generation process. Expanding upon prior studies, we develop a framework that formalizes divisibility in the context of stochastic matrices. We provide a sufficient condition for divisibility that is shown to be necessary in dimension $n=3$, while for $n=2$, all stochastic matrices are shown to be divisible. Using these results, we construct generating sets for dimensions 2 and 3 by specifying the indivisible elements, and, importantly, we give an upper bound for the number of factors required from the generating set to produce the entire semigroup.