On matrix powers in max-algebra

Let A=(aij)∈R¯n×n,N={1,…,n} and DA be the digraph(N,{(i,j);aij>-∞}).(N,{(i,j);aij>-∞}).The matrix A is called irreducible if DA 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)⊗ 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.