由非正交Toeplitz矩阵与秩1扰动生成的特征值和伪谱过程
Eigenvalue and pseudospectrum processes generated by nonnormal Toeplitz matrices with rank 1 perturbations
AI总结：
研究探讨了非正交Toeplitz矩阵与秩1扰动生成的特征值和伪谱过程，揭示了精确特征值与伪谱退化特征值的分离特性，对数值分析具有重要意义。
摘要：
我们引入了由非正交Toeplitz矩阵与加性秩1扰动δJ生成的两种矩阵值动态过程，其中δ∈C，J是全1矩阵。对于每种过程，首先报告了数值获得的特征值的复杂运动。然后推导出确定非零简单特征值运动的具体方程，并阐明零特征值λ₀=0的退化性的时间依赖性。与该方程的比较表明，数值观察到的非零特征值围绕λ₀分布的是原系统之外的精确特征值，而非受计算机未受控舍入误差扰动的系统。随机扰动系统的特征值所分布的复域被识别为包含原系统δJ的伪谱。我们利用对应的非正交Toeplitz算子的符号曲线特征化伪谱过程。在我们的第二个模型中，报告了新现象：在每个时间点，从符号曲线中切出的最外层闭合简单曲线代表精确特征值，而符号曲线内部分则缩小并嵌入包含λ₀的伪谱中。这种精确简单特征值与伪谱关联的退化特征值的分离对数值分析具有意义，因为前者稳定且鲁棒，而后者对扰动高度敏感且不稳定。本研究将与量子物理中非厄米特系统伪谱方法相关。
英文摘要：
We introduce two kinds of matrix-valued dynamical processes generated by nonnormal Toeplitz matrices with the additive rank 1 perturbations $δJ$, where $δ\in {\mathbb{C}}$ and $J$ is the all-ones matrix. For each process, first we report the complicated motion of the numerically obtained eigenvalues. Then we derive the specific equation which determines the motion of non-zero simple eigenvalues and clarifies the time-dependence of degeneracy of the zero-eigenvalue $λ_0=0$. Comparison with the solutions of this equation, it is concluded that the numerically observed non-zero eigenvalues distributing around $λ_0$ are the exact eigenvalues not of the original system, but of the system perturbed by uncontrolled rounding errors of computer. The complex domain in which the eigenvalues of randomly perturbed system are distributed is identified with the pseudospectrum including $λ_0$ of the original system with $δJ$. We characterize the pseudospectrum processes using the symbol curves of the corresponding nonnormal Toeplitz operators without $δJ$. We report new phenomena in our second model such that at each time the outermost closed simple curve cut out from the symbol curve is realized as the exact eigenvalues, but the inner part of symbol curve is reduced in size and embedded in the pseudospectrum including $λ_0$. Such separation of exact simple eigenvalues and a degenerated eigenvalue associated with pseudospectrum will be meaningful for numerical analysis, since the former is stable and robust, but the latter is highly sensitive and unstable with respective to perturbations. The present study will be related to the pseudospectra approaches to non-Hermitian systems developed in quantum physics
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