2009/02/26 by Raphael Loewy, Loewy, Raphael, Bit-Shun Tam +1
Computer Science · Engineering · Mathematics · Physics and Astronomy · #05C50 #15A48 #47A06 #Advanced Mathematical Theories and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Matrix Theory and Algorithms #graph theory and CDMA systems #math.DS #msc:05C50 #msc:15A48 #msc:47A06
paper · pdf · doi:10.48550/arxiv.0902.4589
83 pages, 6 figures
arxiv created 2009/02/26 · openalex publication_date 2009/02/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K be a proper (i.e., closed, pointed, full convex) cone in \Bbb Rn. An n× n matrix A is said to be K-primitive if there exists a positive integer k such that Ak(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.