Matrix semigroups with constant spectral radius

Multiplicative matrix semigroups with constant spectral radius (c.s.r.) are studied and applied to several problems of algebra, combinatorics, functional equations, and dynamical systems. We show that all such semigroups are characterized by means of irreducible ones. Each irreducible c.s.r. semigro...

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Bibliographic Details
Published in:Linear algebra and its applications Vol. 513; pp. 376 - 408
Main Authors: Protasov, V.Yu, Voynov, A.S.
Format: Journal Article
Language:English
Published: Amsterdam Elsevier Inc 15.01.2017
American Elsevier Company, Inc
Subjects:
Combinatorial analysis
Convex body
Ellipsoid
Euclidean space
Finite matrix group
Fractal curves
Functional equations
Joint spectral characteristics
Linear algebra
Linear switching systems
Matrix
Multiplicative semigroup
Polynomial algorithm
Polynomials
Spectral radius
Theorems
Fractal curves
Spectral radius
15B36
Convex body
Finite matrix group
Polynomial algorithm
Linear switching systems
15A30
47D03
Multiplicative semigroup
15A60
Ellipsoid
Joint spectral characteristics
ISSN:0024-3795, 1873-1856
Online Access:Get full text
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Summary:Multiplicative matrix semigroups with constant spectral radius (c.s.r.) are studied and applied to several problems of algebra, combinatorics, functional equations, and dynamical systems. We show that all such semigroups are characterized by means of irreducible ones. Each irreducible c.s.r. semigroup defines walks on Euclidean sphere, all its nonsingular elements are similar (in the same basis) to orthogonal. We classify all nonnegative c.s.r. semigroups and arbitrary low-dimensional semigroups. For higher dimensions, we describe five classes and leave an open problem on completeness of that list. The problem of algorithmic recognition of c.s.r. property is proved to be polynomially solvable for irreducible semigroups and undecidable for reducible ones.
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ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2016.10.013