CSR expansions of matrix powers in max algebra

TL;DR

This paper investigates the behavior of max-algebraic powers of reducible matrices, providing a new expansion method for large powers, analyzing their ultimate behavior, and applying it to determine periodicity efficiently.

Contribution

It introduces a max-algebraic expansion of matrix powers for t > 3n^2, enabling efficient analysis of their long-term behavior and periodicity.

Findings

Matrix powers can be expanded as CS^tR for large t

The expansion can be computed in O(n^4 log n) time

Ultimate behavior of powers reveals periodicity properties

Abstract

We study the behavior of max-algebraic powers of a reducible nonnegative n by n matrix A. We show that for t>3n^2, the powers A^t can be expanded in max-algebraic powers of the form CS^tR, where C and R are extracted from columns and rows of certain Kleene stars and S is diadonally similar to a Boolean matrix. We study the properties of individual terms and show that all terms, for a given t>3n^2, can be found in O(n^4 log n) operations. We show that the powers have a well-defined ultimate behavior, where certain terms are totally or partially suppressed, thus leading to ultimate CS^tR terms and the corresponding ultimate expansion. We apply this expansion to the question whether {A^ty, t>0} is ultimately linear periodic for each starting vector y, showing that this question can be also answered in O(n^4 log n) time. We give examples illustrating our main results.