On primitivity of sets of matrices

• Vincent D. Blondel ^[1] ; Raphaël M. Jungers ^[1] ; Alex Olshevsky ^[2]
  1. [1] Université Catholique de Louvain

     Université Catholique de Louvain

     Arrondissement de Nivelles, Bélgica

     
  2. [2] University of Illinois at Urbana Champaign

     University of Illinois at Urbana Champaign

     Township of Cunningham, Estados Unidos

     
• Localización: Automatica: A journal of IFAC the International Federation of Automatic Control, ISSN 0005-1098, Vol. 61, 2015, págs. 80-88
• Idioma: inglés
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• Resumen

  • A nonnegative matrix AA is called primitive if AkAk is positive for some integer k>0k>0. A generalization by Protasov and Voynov (2012) of this concept to finite sets of matrices is as follows: a set of matrices M={A1,A2,…,Am}M={A1,A2,…,Am} is primitive if Ai1Ai2…AikAi1Ai2…Aik is positive for some indices i1,i2,...,iki1,i2,...,ik. The concept of primitive sets of matrices comes up in a number of problems within the study of discrete-time switched systems. In this paper, we analyze the computational complexity of deciding if a given set of matrices is primitive and we derive bounds on the length of the shortest positive product.

    We show that while primitivity is algorithmically decidable, unless P=NPP=NP it is not possible to decide primitivity of a matrix set in polynomial time. Moreover, we show that the length of the shortest positive sequence can be superpolynomial in the dimension of the matrices. On the other hand, defining PP to be the set of matrices with no zero rows or columns, we give a combinatorial proof of the Protasov–Voynov characterization (2012) of primitivity for matrices in PP which can be tested in polynomial time. This latter observation is related to the well-known 1964 conjecture of Černý on synchronizing automata; in fact, any bound on the minimal length of a synchronizing word for synchronizing automata immediately translates into a bound on the length of the shortest positive product of a primitive set of matrices in PP. In particular, any primitive set of n×nn×n matrices in PP has a positive product of length O(n3)O(n3).