Strong Shift Equivalence and Positive Doubly Stochastic Matrices

TL;DR

This paper explores the relationships between positive stochastic matrices and positive doubly stochastic matrices, establishing conditions for strong shift equivalence and analyzing spectral properties and equivalence classes.

Contribution

It provides new conditions for strong shift equivalence to doubly stochastic matrices and characterizes spectral sets and equivalence class finiteness for primitive matrices.

Findings

Every positive stochastic matrix is strong shift equivalent to a positive doubly stochastic matrix.

The spectra of primitive stochastic and doubly stochastic matrices are identical.

Finiteness of SSE-$R_+$ classes for matrices with positive trace.

Abstract

We give sufficient conditions for a positive stochastic matrix to be similar and strong shift equivalent over $\mathbb{R}_+$ to a positive doubly stochastic matrix through matrices of the same size. We also prove that every positive stochastic matrix is strong shift equivalent over $\mathbb{R}_+$ to a positive doubly stochastic matrix. Consequently, the set of nonzero spectra of primitive stochastic matrices over $\mathbb{R}$ with positive trace and the set of nonzero spectra of positive doubly stochastic matrices over $\mathbb{R}$ are identical. We exhibit a class of $2\times 2$ matrices, pairwise strong shift equivalent over $\mathbb R_+$ through $2\times 2$ matrices, for which there is no uniform upper bound on the minimum lag of a strong shift equivalence through matrices of bounded size. In contrast, we show for any $n\times n$ primitive matrix of positive trace that the set of positive $n\times n$ matrices similar to it contains only finitely many SSE-$\mathbb R_+$ classes.