发表机构
National Sun Yat-sen University; University of North Carolina at Chapel Hill; The Chinese University of Hong Kong, Shenzhen; Chalmers University of Technology and University of Gothenburg（国立中山大学; 北卡罗来纳大学教堂山分校; 香港中文大学（深圳）; 查尔姆斯理工大学和哥德堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对扇区矩阵，推导了使$I+M\boldsymbol{\triangle}$失秩的实角稳定半径公式，证明其通常与复角稳定半径不同，并可通过二维子空间压缩将高阶矩阵结果简化为二阶矩阵结果。
AI 中文摘要
实稳定半径问题是寻找使给定矩阵失秩的最小实扰动，它是对应复稳定半径问题的结构化版本。本文针对扇区矩阵研究该问题的角版本，其中扰动的大小由其最大相位而非最大奇异值衡量。更具体地，对于扇区矩阵$M$，我们研究使$I+M\boldsymbol{\triangle}$失秩的实扇区扰动$\boldsymbol{\triangle}$所需的最小最大相位。我们证明，对于部分矩阵，复角稳定半径与实角稳定半径相同，但通常二者不同，例如当$M$为对角矩阵或复对称矩阵时。对于这些情形，我们推导了实角稳定半径的公式，该公式依赖于$M$的两个最大相位。此外，在这些情形下，我们证明可通过将任意实失稳扰动压缩至由失稳向量的实部和虚部张成的实二维子空间，将$n \times n$矩阵的结果简化为$2 \times 2$矩阵的结果。
英文摘要
The real stability radius problem asks for the smallest real perturbation that makes a given matrix lose rank, and it is a structured counterpart of the corresponding complex stability radius problem. In this paper, we consider an angular version of this question for sectorial matrices, where the size of a perturbation is measured by its largest phase instead of its largest singular value. More precisely, for a sectorial matrix $M$, we investigate the minimum required largest phase of a real and sectorial perturbation $Δ$ for which $I+MΔ$ loses rank. We show that for some matrices, the complex and the real angular stability radius are the same, but that they are in general different, such as when $M$ is diagonal or complex symmetric. For these cases, we derive a formula for the real angular stability radius, and this formula depends on the two largest phases of $M$. Moreover, in these cases we show that the result for $n \times n$ matrices can be reduced to that for $2 \times 2$ matrices by compressing an arbitrary real destabilizing perturbation to the real two-dimensional subspace spanned by the real and imaginary parts of a destabilizing vector.
Comments22 pages
