秩一乘法映射的互补性性质

中文摘要 AI 辅助

本文考虑了与乘法映射 $M_A:\mathbb{S}^n\to\mathbb{S}^n$（定义为 $M_A(X)=AXA^T$）相关的半定线性互补问题，其中 $A\in\mathbb{R}^{n\times n}$ 的秩为一。若相关互补问题的每个可行实例均存在互补解，则称该线性映射具有 $Q_0$-性质。我们证明了两个尖锐且互补的结果。当 $A=xy^T$ 且 $x,y\in\mathbb{R}^n$ 线性无关时，我们构造了一个矩阵 $Q\in\mathbb{S}^n$，使得相关的半定互补问题可行但无互补解，从而证明 $M_A$ 不具有 $Q_0$-性质。反之，当 $A=uu^T$ 且 $u\in\mathbb{R}^n$ 非零时，我们证明 $M_A$ 具有 $Q_0$-性质。这两个结果完整刻画了秩一乘法映射族中的 $Q_0$-性质。

英文摘要

The semidefinite linear complementarity problem associated with the multiplicative map $M_A:\mathbb{S}^n\to\mathbb{S}^n$ defined by $M_A(X)=AXA^T$, is considered, in the case when $A\in\mathbb{R}^{n\times n}$ has rank one. A linear map is said to have the $Q_0$-property if every feasible instance of the associated complementarity problem admits a complementary solution. We prove two sharp, complementary results. When $A=xy^T$ with $x,y\in\mathbb{R}^n$ \emph{linearly independent}, we construct a matrix $Q\in\mathbb{S}^n$ for which the associated semidefinite complementarity problem is feasible, but possesses no complementary solution; proving that $M_A$ \emph{does not have} the $Q_0$-property. Conversely, when $A=uu^T$ for some nonzero $u\in\mathbb{R}^n$, we prove that $M_A$ \emph{has} the $Q_0$-property. These two results provide a complete characterization of the $Q_0$-property within the family of rank-one multiplicative maps.

发表机构

• Indian Institute of Technology Madras（马德拉斯印度理工学院）

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