In the current work, we study the eigenvalue distribution results of a class of non-normal matrix-sequences which may be viewed as a low rank perturbation, depending on a parameter $\beta>1$, of the basic Toeplitz matrix-sequence $\{T_n(e^{\mathbf{i}\theta})\}_{n\in\mathbb{N}}$, $\mathbf{i}^2=-1$. The latter of which has obviously all eigenvalues equal to zero for any matrix order $n$, while for the matrix-sequence under consideration we will show a strong clustering on the complex unit circle. A detailed discussion on the outliers is also provided. The problem appears mathematically innocent, but it is indeed quite challenging since all the classical machinery for deducing the eigenvalue clustering does not cover the considered case. In the derivations, we resort to a trick used for the spectral analysis of the Google matrix plus several tools from complex analysis. We only mention that the problem is not an academic curiosity and in fact stems from problems in dynamical systems and number theory. Additionally, we also provide numerical experiments in high precision, a distribution analysis in the Weyl sense concerning both eigenvalues and singular values is given, and more results are sketched for the limit case of $\beta=1$

翻译：本文研究一类非正规矩阵序列的特征值分布结果，该类序列可视为参数 $\beta>1$ 下对基本托普利茨矩阵序列 $\{T_n(e^{\mathbf{i}\theta})\}_{n\in\mathbb{N}}$（其中 $\mathbf{i}^2=-1$）的低秩扰动。后者显然对所有矩阵阶数 $n$ 的特征值均为零，而本文所考虑的矩阵序列将在复单位圆上呈现强聚类现象，并给出了离群值的详细讨论。该问题看似数学上简单，实则极具挑战性，因为所有用于推导特征值聚类的经典方法均不适用于当前情形。在推导过程中，我们借鉴了Google矩阵谱分析中使用的技巧以及复分析的若干工具。需指出的是，该问题并非纯粹的学术好奇心，实际上源于动力系统与数论中的问题。此外，我们还提供了高精度的数值实验，给出了关于特征值与奇异值的Weyl意义下的分布分析，并对 $\beta=1$ 的极限情形进行了结果概略阐述。