Path Methods for Strong Shift Equivalence of Positive Matrices

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 \(\mathcal{U}\) of ℝ. This paper gives a detailed, unified and generalized presentation of these path methods. New arguments which address arbitrary dense subrings \(\mathcal{U}\) of ℝ are used to show that for any dense subring \(\mathcal{U}\) of ℝ, positive matrices over \(\mathcal{U}\) which have just one nonzero eigenvalue and which are strong shift equivalent over \(\mathcal{U}\) must be strong shift equivalent over \(\mathcal{U}_{+}\). In addition, we show matrices on a path of positive shift equivalent real matrices are SSE over ℝ+; positive rational matrices which are SSE over ℝ+ must be SSE over ℚ+; and for any dense subring \(\mathcal{U}\) of ℝ, within the set of positive matrices over \(\mathcal{U}\) which are conjugate over \(\mathcal{U}\) to a given matrix, there are only finitely many SSE-\(\mathcal{U}_{+}\) classes.