On the core of a cone-preserving map

Abstract

This is the third of a sequence of papers in an attempt to study the Perron-Frobenius theory of a nonnegative matrix and its generalizations from the cone-theoretic viewpoint. Our main object of interest here is the core of a cone-preserving map. If A is an n × n n \times n real matrix which leaves invariant a proper cone K in R n {\mathbb {R}^n} , then by the core of A relative to K, denoted by core K ( A ) {\text {core}}_K(A) , we mean the convex cone ⋂ i = 1 ∞ A i K \bigcap \nolimits _{i = 1}^\infty {{A^i}K} . It is shown that when core K ( A ) {\text {core}}_K(A) is polyhedral, which is the case whenever K is, then core K ( A ) {\text {core}}_K(A) is generated by the distinguished eigenvectors of positive powers of A. The important concept of a distinguished A-invariant face is introduced, which corresponds to the concept of a distinguished class in the nonnegative matrix case. We prove a significant theorem which describes a one-to-one correspondence between the distinguished A-invariant faces of K and the cycles of the permutation induced by A on the extreme rays of core K ( A ) {\text {core}}_K(A) , provided that the latter cone is nonzero, simplicial. By an interplay between cone-theoretic and graph-theoretic ideas, the extreme rays of the core of a nonnegative matrix are fully described. Characterizations of K-irreducibility or A-primitivity of A are also found in terms of core K ( A ) {\text {core}}_K(A) . Several equivalent conditions are also given on a matrix with an invariant proper cone so that its spectral radius is an eigenvalue of index one. An equivalent condition in terms of the peripheral spectrum is also found on a real matrix A with the Perron-Schaefer condition for which there exists a proper invariant cone K suchthat core K ( A ) {\text {core}}_K(A) is polyhedral, simplicial, or a single ray. A method of producing a large class of invariant proper cones for a matrix with the Perron-Schaefer condition is also offered.