Two cores of a nonnegative matrix

TL;DR

This paper investigates the periodicity of eigencones of nonnegative matrices' powers and unifies the spectral theory of matrix powers and cores in max algebra and nonnegative linear algebra.

Contribution

It proves the periodicity of eigencones in both algebraic settings and characterizes the core as a Minkowski sum of eigencones, unifying the spectral theories.

Findings

Eigencones of matrix powers are periodic in both algebras.

The Minkowski sum of eigencones equals the matrix core.

The set of extremal rays of the core is described.

Abstract

We prove that the sequence of eigencones (i.e., cones of nonnegative eigenvectors) of positive powers A^k of a nonnegative square matrix A is periodic both in max algebra and in nonnegative linear algebra. Using an argument of Pullman, we also show that the Minkowski sum of the eigencones of powers of A is equal to the core of A defined as the intersection of nonnegative column spans of matrix powers, also in max algebra. Based on this, we describe the set of extremal rays of the core. The spectral theory of matrix powers and the theory of matrix core is developed in max algebra and in nonnegative linear algebra simultaneously wherever possible, in order to unify and compare both versions of the same theory.