AI 中文总结
本文通过分析噪声矩阵与特征向量的相互作用，提出区域稳定性结果和双跳论证，推导出优于Weyl不等式的新特征值扰动界，适用于对称、Hermitian及矩形矩阵。
AI 中文摘要
设 $A$ 为 $n\times n$ 对称矩阵，其特征值为 $\lambda_1\geq\cdots\geq\lambda_n$。令 $\tilde A:=A+E$，其中 $E$ 为对称噪声矩阵，并将 $\tilde A$ 的特征值记为 $\tilde\lambda_1\geq\cdots\geq\tilde\lambda_n$。对扰动 $|\tilde \lambda_i -\lambda_i|$ 进行界定是线性代数和数值分析中的核心问题。本文通过探索 $E$ 与 $A$ 的特征向量之间的实际相互作用，证明了新的扰动结果。在 $E$ 不针对这些向量进行对抗性作用（例如，若 $E$ 为随机矩阵）的情况下，我们获得了对 Weyl 不等式的显著改进。这些结果可以常规地推广到 Hermitian 和矩形情形。我们作为区域稳定性结果的推论得到这些新界，该区域稳定性结果为实轴上某个区域在扰动后保持稳定（包含相同数量的特征值）提供了充分条件。这一结果具有独立的意义。我们使用围道积分分析证明了区域稳定性结果。这里的主要新技术成分是“双跳”论证，该论证具有鲁棒性，并可能适用于许多涉及 Neumann 级数的其他情形。
英文摘要
Let $A$ be an $n\times n$ symmetric matrix with eigenvalues $λ_1\geq\cdots\geqλ_n$. Let $\tilde A:=A+E$, where $E$ is a symmetric noise matrix, and denote the eigenvalues of $\tilde A$ by $\tildeλ_1\geq\cdots\geq\tildeλ_n$. Bounding the perturbation $|\tilde λ_i -λ_i| $ is a central problem in linear algebra and numerical analysis. In this paper, we prove new perturbation results by exploring the actual interaction between $E$ and the eigenvectors of $A$. In the setting where $E$ does not act adversarially with respect to these vectors (for instance, if $E$ is random), we obtain a considerable improvement over Weyl's inequality. One can routinely extend these results to the Hermitian and rectangular settings. We obtain the new bounds as corollaries of a regional stability result, which provides a sufficient condition for a region on the real line to be stable (containing the same number of eigenvalues) after the perturbation. This result is of independent interest. We prove our regional stability result using contour integral analysis. Our main new technical ingredient here is the \textit{double-jump} argument, which is robust and could be useful in many other situations involving Neumann series.
