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    Barabanov norms, Lipschitz continuity and monotonicity for the max algebraic joint spectral radius


    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:

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