On matrix powers in max-algebra

Let A = (a(ij)) is an element of (R) over bar (nxn), N = {1,...,n} and D-A be the digraph (N, {(i, j): a(ij) > - infinity}). The matrix A is called irreducible if D-A is strongly connected, and strongly irreducible if every max-algebraic power of A is irreducible. A is called robust if for every x with at least one finite component, A((k)) circle times x is an eigenvector of A for some natural number k. We study the eigenvalue-eigenvector problem for powers of irreducible matrices. This enables us to characterise robust irreducible matrices. In particular, robust strongly irreducible matrices are described in terms of eigenspaces of matrix powers. (c) 2006 Elsevier Inc. All rights reserved.