发表机构
Aalto University（阿尔托大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究实数矩阵在 Frobenius 距离下到正规矩阵或特征值在闭左半平面的矩阵集合的最近点是否可为实数，给出否定答案，并证明在维度≥3时实数与复数距离之比无界，且实数输入无实数最小化器的概率严格介于0和1之间。
AI 中文摘要
给定一个实数方阵 $A$ 和一个非空闭的复矩阵目标集 $\u03bcmathcal{E}$，该集合在复共轭下不变且包含实数矩阵，那么在 Frobenius 距离下，$\u03bcmathcal{E}$ 中距离 $A$ 最近的矩阵子集是否总是包含一个实数矩阵？当 $\u03bcmathcal{E}$ 是正规矩阵集合（解决了 N. Higham 提出的问题）或特征值位于闭左半平面的矩阵集合（解决了第一作者和 F. Poloni 提出的问题）时，我们给出否定答案。对于后一个问题，我们还论证了在每一维度 $n\geq 3$ 下，实数距离与复数距离之比是无界的，并且这适用于由酉不变范数诱导的任何距离。对于这两个问题且 $n \geq 3$，我们证明从单位球面 $\\|A\\|_F=1$ 上均匀抽取的实数输入 $A$，在 $\u03bcmathcal{E}$ 上以严格介于 0 和 1 之间的概率没有实数最小化器。本文还补充了一些适用于更一般目标集 $\u03bcmathcal{E}$ 的进一步结果。
英文摘要
Given a real square matrix $A$ and a nonempty closed target set of complex matrices $\mathcal E$, invariant by complex conjugation and containing real matrices, does the subset of $\mathcal{E}$ consisting of the matrices nearest to $A$ in the Frobenius distance always contain a real matrix? We give negative answers when $\mathcal{E}$ is either the set of normal matrices (solving a question posed by N. Higham) or the set of matrices whose eigenvalues lie in the closed left half-plane (solving a question posed by the first author and F. Poloni). For the latter problem, we also argue that the ratio between the real and complex distances is unbounded in every dimension $n\geq 3$, and this holds for every distance induced by a unitarily invariant norm. For both problems and $n \geq 3$, we show that a real input $A$ uniformly drawn from the unit sphere $\|A\|_F=1$ has no real minimizer over $\mathcal{E}$ with probability strictly between $0$ and $1$. The paper is complemented by some further results that are valid for more general target sets $\mathcal{E}$.
