AI 中文总结
本文针对余正优化领域中保余正线性映射刻画的问题，提出仅用序理论论证的新方法，得到Riesz*同态的表示定理，统一了相关成果并修正了标准线性映射保余正性的刻画。
AI 中文摘要
对于锥$K\subseteq \mathbb{R}^n$，若对所有$x\in K$都有$x^\top A x\geq 0$，则实对称矩阵$A$称为$K$-余正矩阵。这类矩阵在余正优化和线性互补问题中处于核心地位。然而，即使对于$K:=\mathbb{R}^n_+$，保$K$-余正锥的线性映射的完整刻画仍未解决。本文开发了一种仅使用序理论论证的新方法来研究余正保序映射，考虑了一类更小的余正保序映射，称为Riesz*同态，并开发了一种通用技术，可直接从$S_n$上Riesz*同态的表示定理推导这类保序映射的结构。本文的主要成果如下：1）获得了所有实对称矩阵构成的偏序向量空间上Riesz*同态的表示定理，该空间赋予所有$K$-余正矩阵构成的锥；2）作为该表示定理的推论，本文重现了[Shitov, Proc. Amer. Math. Soc., 2021]和[Gowda et al., Linear Alg. Appl., 2013]的主要结果，为研究锥自同构提供了统一框架；3）引入了$(K_1,K_2)$无符号矩阵$P\in M_{m\times n}$的概念，其由代数条件$P[K_1]\subseteq K_2\cup (-K_2)$定义，其中$K_1\subseteq \mathbb{R}^n$、$K_2\subseteq \mathbb{R}^m$为锥，本文也给出了这类矩阵的刻画；4）证明了标准形式的线性映射（$A\mapsto P^\top AP$，其中$P\in M_{m\times n}$）保余正性当且仅当$P$是$(K_2,K_1)$无符号矩阵，修正了近期对这类保$\mathbb{R}^n_+$-余正性的映射的刻画。
英文摘要
For a cone $K\subseteq \mathbb{R}^n$, a real symmetric matrix $A$ is called $K$-copositive if $x^\top A x\geq 0$ for every $x\in K.$ This class of matrices plays a central role in copositive optimization and linear complementarity problems. However, a complete characterization of linear maps that preserve the $K$-copositive cone is unknown, even for $K:=\mathbb{R}^n_+$. In this paper, we develop a new approach to copositivity preservers that uses only order-theoretic arguments. We consider a smaller class of copositivity preservers, called Riesz* homomorphisms, and develop a general technique to deduce the structure of these preservers directly from a representation theorem of Riesz* homomorphisms on $S_n$. Following are the main outcomes of this paper: 1) We obtain a representation theorem for Riesz* homomorphisms on the partially ordered vector space of all real symmetric matrices endowed with the cone of all $K$-copositive matrices. 2) As a corollary of our representation theorem, we recover the main results of [Shitov, Proc. Amer. Math. Soc., 2021] and [Gowda et al., Linear Alg. Appl., 2013], providing a unified framework for studying cone automorphisms. 3) We introduce the notion of a $(K_1,K_2)$-unisigned matrix $P\in M_{m\times n}$, defined by the algebraic condition $P[K_1]\subseteq K_2\cup (-K_2)$, for cones $K_1\subseteq \mathbb{R}^n$ and $K_2\subseteq \mathbb{R}^m$. We also provide a characterization of such matrices. 4) We prove that a linear map of the standard form ($A\mapsto P^\top AP$; for $P\in M_{m\times n}$) preserves copositivity if and only if $P$ is $(K_2,K_1)$-unisigned, correcting a recent characterization of such maps preserving the $\mathbb{R}^n_+$-copositivity.
