Maximal exponents of polyhedral cones (I)

doi:10.1016/j.jmaa.2009.11.016

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Title:	Maximal exponents of polyhedral cones (I)
Authors:	Loewy, Raphael;Tam, Bit-Shun
Contributors:	淡江大學數學學系
Keywords:	Cone-preserving map;K-primitive matrix;Exponents;Polyhedral cone
Date:	2010-05
Issue Date:	2011-10-01 21:08:22 (UTC+8)
Publisher:	Maryland Heights: Academic Press
Abstract:	Let K be a proper (i.e., closed, pointed, full convex) cone in 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 called the exponent of K and is denoted by γ(K). It is proved that if K is an n-dimensional polyhedral cone with m extreme rays then for any K-primitive matrix A, γ(A) ≦ (mA − 1)(m − 1) + 1, where mA denotes the degree of the minimal polynomial of A, and the equality holds only if the digraph (E,P(A,K)) associated with A (as a cone-preserving map) is equal to the unique (up to isomorphism) usual digraph associated with an m x m primitive matrix whose exponent attains Wielandt’s classical sharp bound. As a consequence, for any n-dimensional polyhedral cone K with m extreme rays, γ(K) ≦ (n−1)(m−1)+1. Our work answers in the affirmative a conjecture posed by Steve Kirkland about an upper bound of γ(K) for a polyhedral cone K with a given number of extreme rays.
Relation:	Journal of Mathematical Analysis and Applications 365(2), pp.570-583
DOI:	10.1016/j.jmaa.2009.11.016
Appears in Collections:	[Department of Applied Mathematics and Data Science] Journal Article

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