Max algebraic powers of irreducible matrices in the periodic regime: An application of cyclic classes

https://doi.org/10.1016/j.laa.2009.04.027Get rights and content
Under an Elsevier user license
open archive

Abstract

In max algebra it is well known that the sequence of max algebraic powers Ak, with A an irreducible square matrix, becomes periodic after a finite transient time T(A), and the ultimate period γ is equal to the cyclicity of the critical graph of A.

In this connection, we study computational complexity of the following problems: (1) for a given k, compute a periodic power Ar with rk(modγ) and rT(A), (2) for a given x, find the ultimate period of {Alx}. We show that both problems can be solved by matrix squaring in O(n3logn) operations. The main idea is to apply an appropriate diagonal similarity scaling AX-1AX, called visualization scaling, and to study the role of cyclic classes of the critical graph.

AMS classification

Primary: 15A48, 15A06
Secondary: 06F15

Keywords

Max-plus algebra
Tropical algebra
Diagonal similarity
Cyclicity
Imprimitive matrix

Cited by (0)

This research was supported by EPSRC Grant RRAH12809 and RFBR Grant 08-01-00601.