Path methods for strong shift equivalence of positive matrices

TL;DR

This paper generalizes path methods for establishing strong shift equivalence of positive matrices, proving new results about their classification over various dense subrings of the real numbers.

Contribution

It provides a unified framework for path methods, extending results to arbitrary dense subrings and establishing finiteness and equivalence properties of positive matrices.

Findings

Positive matrices with one nonzero eigenvalue are SSE over U_+ if they are SSE over U.

Positive real matrices on a path of shift equivalent matrices are SSE over R_+.

Finitely many SSE-U_+ classes exist within conjugate positive matrices over U.

Abstract

In the early 1990's, Kim and Roush developed path methods for establishing strong shift equivalence (SSE) of positive matrices over a dense subring U of the real numbers R. This paper gives a detailed, unified and generalized presentation of these path methods. New arguments which address arbitrary dense subrings U of R are used to show that for any dense subring U of R, positive matrices over U which have just one nonzero eigenvalue and which are strong shift equivalent over U must be strong shift equivalent over U_+. In addition, we show positive real matrices on a path of shift equivalent positive real matrices are SSE over R_+; positive rational matrices which are SSE over R_+ must be SSE over Q_+; and for any dense subring U of R, within the set of positive matrices over U which are conjugate over U to a given matrix, there are only finitely many SSE-U_+ classes.