Sets of Nonnegative Matrices with Positive Inhomogeneous Products*

Let X be a set of k x 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. A matrix A = ( aij) with real elements is called nonnegative (A > 0) if aij> 0, and is called positive (A > 0) if aij> 0 for all i,j. A primitive matrix A is defined to be a k X k nonnegative matrix (1 < k < 00) such that, for some positive integer n, A” > 0. Primitive matrices share important spectral and contractive properties with positive matrices and have been studied extensively [ 11. A primitive matrix A has index g if Ag > 0 but none of the matrices A”, 0 < n < g, is positive. Wielandt [13] states without proof that, for any primitive matrix A, if g(A) is the index of A, then g(A) < k2 2k +2, (1) *In honor of Mark Kac. LINEAR ALGEBRA AND ITS APPLICATIONS 47~185-192 (1982) 185 0 Elsevier Science Publishing Co., Inc., 1982 52 Vanderbilt Ave., New York, NY 10017 00243795/82/06018548$02.75 186 JOEL E. COHEN AND PETER H. SELLERS and the upper bound is attained by g(B), where B= 0 1 0 .*. 0 0 0 1 ... 0 0 0 0 ... 1 1 1 0 .** 0 \