Sets of nonnegative matrices without positive productsстатья
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Дата последнего поиска статьи во внешних источниках: 18 июля 2013 г.
Аннотация:For an arbitrary irreducible set of nonnegative d × 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 ofRd, onwhich all given matrices
act as permutations.Moreover, there always exists a unique maximal
partition (with the maximal number of parts) possessing this property,
andthenumber ofparts is expressedby 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.