2009/09/15 by Ian D. Morris, Morris, Ian D.
Computer Science · Mathematics · #15A18 #15A60 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.0909.2800
openalex publication_date 2009/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A set of matrices is said to have the finiteness property if the maximal rate of exponential growth of long products of matrices drawn from that set is realised by a periodic product. The extent to which the finiteness property is prevalent among finite sets of matrices is the subject of ongoing research. In this article we give a condition on a finite irreducible set of matrices which guarantees that the finiteness property holds not only for that set, but also for all sufficiently nearby sets of equal cardinality. We also prove a theorem giving conditions under which the Barabanov norm associated to a finite irreducible set of matrices is unique up to multiplication by a scalar, and show that in certain cases these conditions are also persistent under small perturbations.