Result: Sets of nonnegative matrices without positive products

Title:
Sets of nonnegative matrices without positive products
Authors:
PROTASOV, V. Yu, VOYNOV, A. S
Source:
Linear algebra and its applications. 437(3):749-765
Publisher Information:
Amsterdam: Elsevier, 2012.
Publication Year:
2012
Physical Description:
print, 36 ref
Original Material:
INIST-CNRS
Subject Terms:
Mathematics, Mathématiques, Sciences exactes et technologie, Exact sciences and technology, Sciences et techniques communes, Sciences and techniques of general use, Mathematiques, Mathematics, Combinatoire. Structures ordonnées, Combinatorics. Ordered structures, Combinatoire, Combinatorics, Problèmes combinatoires classiques, Classical combinatorial problems, Algèbre, Algebra, Algèbre linéaire et multilinéaire, matrices, Linear and multilinear algebra, matrix theory, Chaîne Markov, Markov chain, Cadena Markov, Combinatoire énumérative, Enumerative combinatorics, Complexité calcul, Computational complexity, Complejidad computación, Echange désordonné, Scrambling, Intercambio desordendo, Exposant Lyapunov, Lyapunov exponent, Exponente Lyapunov, Matrice non négative, Non negative matrix, Matriz no negativa, Matrice permutation, Permutation matrix, Matriz permutación, Modélisation, Modeling, Modelización, Méthode polynomiale, Polynomial method, Método polinomial, Optimisation combinatoire, Combinatorial optimization, Optimización combinatoria, Opérateur irréductible, Irreducible operator, Operador irreductible, Opérateur positif, Positive operator, Operador positivo, Partition, Partición, Permutation, Permutación, Prise de décision, Decision making, Toma decision, Problème valeur propre, Eigenvalue problem, Problema valor propio, Randomisation, Randomization, Aleatorización, Répétition, Repetition, Repetición, Semigroupe, Semigroup, Semigrupo, Temps polynomial, Polynomial time, Tiempo polinomial, Théorème Frobenius, Frobenius theorem, Teorema Frobenius, Valeur Perron, Perron value, Valor Perron, 05A05, 15B48, 15B51, Multiplicative semigroup, Nonnegative matrix, Polynomial algorithm, Primitivity, Scrambling matrix
Document Type:
Academic journal Article
File Description:
text
Language:
English
Author Affiliations:
Dept. of Mechanics and Mathematics, Moscow State University, Vorobyovy Gory, 119992 Moscow, Russian Federation
ISSN:
0024-3795
Access URL:
http://pascal-francis.inist.fr/vibad/index.php?action=search&terms=25975587
Rights:
Copyright 2015 INIST-CNRS
CC BY 4.0
Sauf mention contraire ci-dessus, le contenu de cette notice bibliographique peut être utilisé dans le cadre d’une licence CC BY 4.0 Inist-CNRS / Unless otherwise stated above, the content of this bibliographic record may be used under a CC BY 4.0 licence by Inist-CNRS / A menos que se haya señalado antes, el contenido de este registro bibliográfico puede ser utilizado al amparo de una licencia CC BY 4.0 Inist-CNRS
Notes:
Mathematics
Accession Number:
edscal.25975587
Database:
PASCAL Archive

Further Information

For an arbitrary irreducible set of nonnegative d x d-matrices, we consider the following problem: does there exist a strictly positive product (with repetitions permitted) of those matrices? Under some general assumptions, we prove that if it does not exist, then there is a partition of the set of basis vectors of ℝd, on which all given matrices act as permutations. Moreover, there always exists a unique maximal partition (with the maximal number of parts) possessing this property, and the number of parts is expressed by eigenvalues of matrices. This generalizes well-known results of Perron-Frobenius theory on primitivity of one matrix to families of matrices. We present a polynomial algorithm to decide the existence of a positive product for a given finite set of matrices and to build the maximal partition. Similar results are obtained for scrambling products. Applications to the study of Lyapunov exponents, inhomogeneous Markov chains, etc. are discussed.