Maximal Exponents of K-Primitive Matrices: The Polyhedral Cone Case

Abstract

Let K be a proper (i.e., closed, pointed, full convex) cone in R n . An n × n matrix A is said to be K-primitive if there exists a positive integer k such that A k (K \ {0}) ⊆ int K; the least such k is referred to as the exponent of A and is denoted by γ(A). For a polyhedral cone K, the maximum value of γ(A), taken over all K-primitive matrices A, is denoted by γ(K). It is proved that for any positive integers m, n, 3 ≤ n ≤ m, the maximum value of γ(K), as K runs through all n-dimensional polyhedral cones with m extreme rays, equals (n − 1)(m − 1) + 1 when m is even or m and n are both odd, and is at least (n − 1)(m − 1) and at most (n−1)(m−1)+1 when m is odd and n is even. For the cases when m = n, m = n+1 or n = 3, the cones K and the corresponding K-primitive matrices A such that γ(K) and γ(A) attain the maximum value are identified up to respectively linear isomorphism and cone-equivalence modulo positive scalar multiplication.