Csr Expansions of Matrix Powers in Max Algebra

We study the behavior of max-algebraic powers of a reducible nonnegative matrix A ∈ Rn×n + . We show that for t ≥ 3n2, the powers At can be expanded in max-algebraic sums of terms of the form CStR, where C and R are extracted from columns and rows of certain Kleene stars, and S is diagonally similar to a Boolean matrix. We study the properties of individual terms and show that all terms, for a given t ≥ 3n2, can be found in O(n4 logn) operations. We show that the powers have a well-defined ultimate behavior, where certain terms are totally or partially suppressed, thus leading to ultimate CStR terms and the corresponding ultimate expansion. We apply this expansion to the question whether {Aty, t ≥ 0} is ultimately linear periodic for each starting vector y, showing that this question can be also answered in O(n4 logn) time. We give examples illustrating our main results.