Strong shift equivalence and shear adjacency of nonnegative square integer matrices (Q1090381)

For two square nonnegative integer matrices A, B, write \(A\approx_ 1B\) if \(A=R\cdot S\) and \(B=S\cdot R\) for some two nonnegative integer matrices R, S. The transitive closure of \(\approx_ 1\) is strong shift equivalence. The author presents a sufficient condition for deciding strong shift equivalence: that A and B are in the same component of a directed graph whose vertices are all \(n\times n\) nonnegative integer matrices sharing a fixed characteristic polynomial and whose edges correspond to certain elementary similarities. This result may be used to determine whether two given topological Markov chains are topologically conjugate. A theorem of \textit{R. F. Williams} [Ann. Math., II. Ser. 98, 120-153 (1973; Zbl 0282.58008)] states a criterion: iff their transition matrices are strongly shift equivalent.