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

TL;DR

This paper determines the maximum exponents of K-primitive matrices for polyhedral cones in real space, revealing exact values and classifications for various dimensions and cone structures.

Contribution

It establishes explicit formulas for the maximum exponents of K-primitive matrices in polyhedral cones and classifies the extremal cases for specific dimensions and configurations.

Findings

Maximum exponent formula for even m or both m and n odd: (n-1)(m-1)+1

Lower and upper bounds for odd m and even n: at least (n-1)(m-1), at most (n-1)(m-1)+1

Classification of cones and matrices attaining maximum exponent for certain cases

Abstract

Let $K$ be a proper (i.e., closed, pointed, full convex) cone in ${\Bbb R}^n$. An $n\times n$ matrix $A$ is said to be $K$-primitive if there exists a positive integer $k$ such that $A^k(K \setminus \{0 \}) \subseteq$ int $K$; the least such $k$ is referred to as the exponent of $A$ and is denoted by $\gamma(A)$. For a polyhedral cone $K$, the maximum value of $\gamma(A)$, taken over all $K$-primitive matrices $A$, is denoted by $\gamma(K)$. It is proved that for any positive integers $m,n, 3 \le n \le m$, the maximum value of $\gamma(K)$, 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 $\gamma(K)$ and $\gamma(A)$ attain the maximum value are identified up to respectively linear isomorphism and cone-equivalence modulo positive scalar multiplication.