A Bound for the Rank-One Transient of Inhomogeneous Matrix Products in Special Case

TL;DR

This paper establishes a bound on the length of inhomogeneous max-plus matrix products after which they become rank-one, providing insights into their transient behavior under specific assumptions.

Contribution

It introduces a bound on the transient length for inhomogeneous max-plus matrix products to become rank-one, extending previous steady-state analyses.

Findings

Bound on the transient length for rank-one behavior

Applicable to matrix products exceeding the bound

Enhances understanding of max-plus algebra dynamics

Abstract

We consider inhomogeneous matrix products over max-plus algebra, where the matrices in the product satisfy certain assumptions under which the matrix products of sufficient length be rank-one, as it was shown in [6][L. Shue, B.D.O. Anderson, S. Dey: On steady state properties of certain max-plus products. Proceedings of the American Control Conference, Philadelphia, Pensylvania, (June 1998), 1909 1913.]. We establish a bound on the transient after which this starts to happen for any product of matrices whose length exceeds that bound.