The maximum number of Pareto eigenvalues of a real matrix of order four is 23

Abstract

For a given real matrix $A\in {\mathbb{R}}^{n\times n}$, a Pareto eigenvalue of $A$ is a real number $\hat{I}»\in \mathbb{R}$ for which there exists a nonzero vector $x\in {\mathbb{R}}^n\setminus \{0\}$ such that $$0\le x\perp Ax-\hat{I}»x\ge 0.$$ We prove that every matrix $A\in {\mathbb{R}}^{4\times 4}$ has at most $23$ distinct Pareto eigenvalues. We first prove the result for matrices of order $4$ satisfying two conditions: every real eigenvalue of a principal submatrix is simple, and two different principal submatrices have no real eigenvalue in common. For these matrices, a fixed point argument shows that the number of Pareto eigenvalues is odd. Previous known results show that the Pareto capacity of order $4$ is between $23$ and $26$. Thus only $25$ remains to exclude. We exclude this case by using the support profiles in order $3$ and identities involving eigenvectors of principal submatrices. A perturbation and fixed point index argument then extends the bound to all real matrices of order $4$. An exact symbolic computation certifies that an explicit matrix of order $4$ has exactly $23$ distinct regular Pareto eigenvalues.