Guglielmi, Nicola and Mason, Oliver and Wirth, Fabian (2018) Barabanov norms, Lipschitz continuity and monotonicity for the max algebraic joint spectral radius. Linear Algebra and its Applications, 550. pp. 37-58. ISSN 0024-3795

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Abstract

We present several results describing the interplay between the max algebraic joint spectral radius (JSR) for compact sets of matrices and suitably defined matrix norms. In particular, we extend a classical result for the conventional algebra, showing that the max algebraic JSR can be described in terms of induced norms of the matrices in the set. We also show that for a set generating an irreducible semigroup (in a cone-theoretic sense), a monotone Barabanov norm always exists. This fact is then used to show that the max algebraic JSR is locally Lipschitz continuous on the space of compact irreducible sets of matrices with respect to the Hausdorff distance. We then prove that the max algebraic JSR is locally Hoelder continuous on the space of compact sets of nonnegative matrices. Finally, we prove a strict monotonicity property for the max algebraic JSR that echoes a fact for the classical JSR. The single matrix characterisation of the max algebraic JSR plays a vital role in our proofs.

Item Type:	Article
Keywords:	Max algebra; Joint spectral radius; Finiteness property; Barabanov norms;
Academic Unit:	Faculty of Science and Engineering > Mathematics and Statistics Faculty of Science and Engineering > Research Institutes > Hamilton Institute
Item ID:	13194
Identification Number:	https://doi.org/10.1016/j.laa.2018.01.042
Depositing User:	Oliver Mason
Date Deposited:	28 Aug 2020 11:47
Journal or Publication Title:	Linear Algebra and its Applications
Publisher:	Elsevier
Refereed:	Yes
URI:	Publisher
Use Licence:	This item is available under a Creative Commons Attribution Non Commercial Share Alike Licence (CC BY-NC-SA). Details of this licence are available here

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