Support profiles of full-capacity Pareto spectra of order three

Abstract

For a given real matrix $A\in\R^{n\times n}$ of order $n\geq 1$, a Pareto eigenvalue is a scalar $\lambda\in\R$ for which there exists a nonzero vector $x\in\R^n_+$ such that $Ax-\lambda x\in\R^n_+$ and $\langle x,Ax-\lambda x\rangle=0$. It is knwon that a real matrix of order $n=3$ can have at most $9$ distinct Pareto eigenvalues. We study the matrices for which this maximal number is attained and classify the way in which the $9$ Pareto eigenvalues are produced by their supports. A Pareto eigenvalue may be produced by more than one support. We therefore choose one producing support for each distinct values. The main contribution of this paper is to prove that, for every such choice, the numbers of values assigned to supports of sizes $1$, $2$, and $3$ are necessarily $$ (1,5,3),\qquad (2,4,3),\qquad\text{or}\qquad (2,5,2). $$ The proof uses bounds from the order $n=2$ problem and several restrictions on singleton, pair, and full supports. In particular, the graph formed by the pair supports producing $2$ values contains no triangle. These arguments also gives anohter proof that the maximum Pareto capacity in order $3$ is equal to $9$. For each of the $3$ profiles, we give an explicit full-capacity matrix with $9$ regular Pareto eigenvalues. We also prove that each profile occurs on a nonempty Euclidean-open set where every Pareto eigenvalue has a unique producing support.