发表机构
Soongsil University; University of Maryland Baltimore County; University of Maryland, College Park（崇实大学; 马里兰大学巴尔的摩分校; 马里兰大学帕克分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文刻画了双曲系统上$e$-双随机矩阵的线性保持算子，揭示了其与优超的联系，并证明了由Jordan框架诱导的矩阵是极端点。
AI 中文摘要
在次数为 $n$ 的双曲（多项式）系统 $(\mathcal{V}, p, e)$ 的设定下，一个 $n$ 元组 $\mathbf{A} = \big[ a_1, a_2, \dots, a_n \big]$ 被称为 $e$-双随机 $\mathcal{V}$-矩阵，如果每个 $a_i$ 属于双曲锥，具有迹 $1$，并且 $a_1 + a_2 + \dots + a_n = e$。在本文中，我们刻画了这类 $\mathcal{V}$-矩阵的线性保持算子，描述了 $e$-双随机性与优超之间的某些联系，并研究了所有 $e$-双随机 $\mathcal{V}$-矩阵集合的极端点。例如，我们证明：（i）当 $n>1$ 时，正的、单位保持且迹保持的变换是 $\mathcal{V}$ 上保持 $e$-双随机性的（唯一的）线性变换；（ii）当 $\mathbf{A}$ 是 $e$-双随机时，线性组合 $\sum_{i=1}^{n} r_ia_i$ 的特征值向量被系数向量 $(r_1, r_2, \dots, r_n)^{T} \in \mathbb{R}^n$ 优超；（iii）当 $p$ 完备时，由（广义）Jordan 框架诱导的 $e$-双随机 $\mathcal{V}$-矩阵是所有 $e$-双随机 $\mathcal{V}$-矩阵的紧凸集的极端点。
英文摘要
In the setting of a hyperbolic (polynomial) system $(\mathcal{V}, p, e)$ of degree $n$, an $n$-tuple $\mathbf{A} = \big[ a_1, a_2, \dots, a_n \big]$ is said to be an $e$-doubly stochastic $\mathcal{V}$-matrix if each $a_i$ belongs to the hyperbolicity cone, has trace $1$, and $a_1 + a_2 + \dots + a_n = e$. In this article, we characterize linear preservers of such $\mathcal{V}$-matrices, describe some connections between $e$-doubly stochasticity and majorization, and study extreme points of the set of all $e$-doubly stochastic $\mathcal{V}$-matrices. We show, for example, that $(i)$ when $n>1$, positive, unital, and trace-preserving transformations are (the only) linear transformations on $\mathcal{V}$ that preserve $e$-doubly stochasticity; $(ii)$ when $\mathbf{A}$ is $e$-doubly stochastic, the eigenvalue vector of the linear combination $\sum_{i=1}^{n} r_ia_i$ is majorized by the coefficient vector $(r_1, r_2, \dots, r_n)^{T} \in \mathbb{R}^n$; and $(iii)$ when $p$ is complete, $e$-doubly stochastic $\mathcal{V}$-matrices induced by (generalized) Jordan frames are extreme points of the compact convex set of all $e$-doubly stochastic $\mathcal{V}$-matrices.
Comments24 pages
