AI 中文总结
研究三阶实矩阵的帕累托特征值，通过对其支撑分类，证明产生\(9\)个帕累托特征值时，支撑大小为\(1\)、\(2\)、\(3\)的值的数量组合为\((1,5,3)\)、\((2,4,3)\)或\((2,5,2)\)，并给出相关矩阵及分布出现的开集。
AI 中文摘要
对于给定的\(n\geq1\)阶实矩阵\(A\in\R^{n\times n}\)，帕累托特征值是一个标量\(\lambda\in\R\)，使得存在非零向量\(x\in\R^n_+\)，满足\(Ax - \lambda x\in\R^n_+\)且\(\langle x,Ax - \lambda x\rangle = 0\)。已知\(n = 3\)阶实矩阵最多有\(9\)个不同的帕累托特征值。本文研究达到此最大数量的矩阵，并根据其支撑对\(9\)个帕累托特征值的产生方式进行分类。一个帕累托特征值可能由多个支撑产生，因此为每个不同值选择一个产生支撑。本文的主要贡献是证明，对于每一种这样的选择，分配给大小为\(1\)、\(2\)和\(3\)的支撑的值的数量必然是\((1,5,3)\)、\((2,4,3)\)或\((2,5,2)\)。证明使用了\(n = 2\)问题的界以及对单元素、对和全支撑的几个限制。特别是，产生\(2\)个值的对支撑形成的图不包含三角形。这些论证还给出了另一个证明，即三阶的最大帕累托容量等于\(9\)。对于这\(3\)种分布中的每一种，我们给出一个具有\(9\)个正则帕累托特征值的显式全容量矩阵。我们还证明，每种分布都出现在一个非空的欧几里得开集上，其中每个帕累托特征值都有一个唯一的产生支撑。
英文摘要
For a given real matrix $A\in\R^{n\times n}$ of order $n\geq 1$, a Pareto eigenvalue is a scalar $λ\in\R$ for which there exists a nonzero vector $x\in\R^n_+$ such that $Ax-λx\in\R^n_+$ and $\langle x,Ax-λx\rangle=0$. 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.
Comments22 pages
