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- W2963669387 abstract "A nonnegative matrix A is called primitive if Ak is positive for some integer k > 0. A generalization of this concept to sets of matrices is as follows: a set of matrices M = {A1,A2, ..., Am} is primitive if Ai1Ai2 ...Aik is positive for some indices i1, i2, ..., ik. The concept of primitive sets of matrices is of importance in several applications, including the problem of computing the Lyapunov exponents of switching 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 = NP it is not possible to decide positivity of a matrix set in polynomial time. Moreover, we show that the length of the shortest positive sequence can be exponential in the dimension of the matrices. On the other hand, we give a simple combinatorial proof of the fact that when the matrices have no zero rows nor zero columns, primitivity can be decided in polynomial time. This latter observation is related to the well-known 1964 conjecture of Cerny on synchronizing automata. Finally, we show that for such matrices the length of the shortest positive sequence is at most polynomial in the dimension." @default.
- W2963669387 created "2019-07-30" @default.
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- W2963669387 date "2013-12-01" @default.
- W2963669387 modified "2023-09-23" @default.
- W2963669387 title "On primitivity of sets of matrices" @default.
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- W2963669387 doi "https://doi.org/10.1109/cdc.2013.6760072" @default.
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