Path components of matrices and strong shift equivalence over Q+

Abstract

We show that two positive matrices over Q + or R + which lie in the same component of a set { MϵM n( R) : M is conjugate over R to A} are strong shift equivalent. It is a simple corollary that 2 × 2 matrices over Q + are strong shift equivalent if they are shift equivalent. Conversely, we prove that if two matrices A, B over R + are strong shift equivalent, then A ⊕ 0, B ⊕ 0 can be joined by a path of nonnegative real matrices within the strong shift equivalence class of A, up to conjugation of B ⊕ 0 by a permutation matrix.