Path Methods for Strong Shift Equivalence of Positive Matrices

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 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 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 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.