This article concerns the spectral analysis of matrix‐sequences which can be written as a non‐Hermitian perturbation of a given Hermitian matrix‐sequence. The main result reads as follows. Suppose that for every n there is a Hermitian matrix X_n of size n and that {X_n}_n∼_λf, that is, the matrix‐sequence {X_n}_n enjoys an asymptotic spectral distribution, in the Weyl sense, described by a Lebesgue measurable function f; if $\Vert Y_n{\Vert }_2=o(\sqrt{n})$ with ‖·‖₂ being the Schatten 2 norm, then {X_n+Y_n}_n∼_λf. In a previous article by Leonid Golinskii and the second author, a similar result was proved, but under the technical restrictive assumption that the involved matrix‐sequences {X_n}_n and {Y_n}_n are uniformly bounded in spectral norm. Nevertheless, the result had a remarkable impact in the analysis of both spectral distribution and clustering of matrix‐sequences arising from various applications, including the numerical approximation of partial differential equations (PDEs) and the preconditioning of PDE discretization matrices. The new result considerably extends the spectral analysis tools provided by the former one, and in fact we are now allowed to analyze linear PDEs with (unbounded) variable coefficients, preconditioned matrix‐sequences, and so forth. A few selected applications are considered, extensive numerical experiments are discussed, and a further conjecture is illustrated at the end of the article.

中文翻译：

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埃尔米特矩阵序列的非埃尔米特扰动及其在偏微分方程数值逼近谱分析中的应用

本文涉及矩阵序列的频谱分析，可以将其写为给定埃尔米特矩阵序列的非埃尔米特扰动。主要结果如下。假设对于每Ñ有一个Hermitian矩阵X _Ñ大小的Ñ和{ X _Ñ } _Ñ〜_λ ˚F，即，矩阵序列{ X _Ñ } _Ñ享有渐近光谱分布，在外尔意义上说，所描述通过Lebesgue可测函数f ; 如果$\Vert {ÿ}_{ñ}{\Vert }_2=Ø（\sqrt{ñ}）$与‖·‖ ₂是所述的Schatten 2范数，则{ X _Ñ + ý _ñ } _Ñ〜_λ ˚F。在列昂尼德·戈林斯基（Leonid Golinskii）和第二作者的上一篇文章中，也证明了类似的结果，但是在技术限制性假设下，涉及的矩阵序列{ X _n } _n和{ Y _n } _n在频谱范数上是一致的 但是，该结果对频谱分布和矩阵序列聚类的分析产生了显着影响，其中包括各种应用，包括偏微分方程（PDE）的数值逼近和PDE离散化矩阵的预处理。新结果大大扩展了前者提供的光谱分析工具，实际上，我们现在可以分析具有（无穷大）可变系数，预处理矩阵序列等的线性PDE。考虑了一些选定的应用程序，讨论了广泛的数值实验，并在文章结尾处说明了进一步的猜想。