Sets of nonnegative matrices with positive inhomogeneous products

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Abstract

Let X be a set of k×k matrices in which each element is nonnegative. For a positive integer n, let P(n) be an arbitrary product of n matrices from X, with any ordering and with repetitions permitted. Define X to be a primitive set if there is a positive integer n such that every P(n) is positive [i.e., every element of every P(n) is positive]. For any primitive set X of matrices, define the index g(X) to be the least positive n such that every P(n) is positive. We show that if X is a primitive set, then g(X)⩽2k−2. Moreover, there exists a primitive set Y such that g(Y) = 2k−2.

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In honor of Mark Kac.