预解式何时类似秩一矩阵？
When is the Resolvent Like a Rank One Matrix?
AI总结：
该研究探讨预解式接近秩一矩阵的复平面点集，建立其与ε-伪谱的关联，推导若尔当块、大型Toeplitz矩阵的相关结果及特征值圆盘结论，还分析了对应最大奇异值的左右奇异向量内积绝对值小于ε的点集性质并讨论应用。
摘要：
对于方阵$A$，其在点$z \in \mathbb{C}$处的预解式定义为$(A-zI )^{-1}$。我们研究满足以下条件的点$z \in \mathbb{C}$构成的集合：预解式与其最近的秩一矩阵之间的2-范数相对差小于给定的$\varepsilon\in (0,1)$。我们建立了该集合与$A$的$\varepsilon$-伪谱之间的关系，并针对若尔当块（Jordan blocks）以及一类大型Toeplitz矩阵推导了关于该集合的具体结果。我们还推导出了包含在该集合内的、围绕$A$的特征值的圆盘，这也带来了关于包含在$A$的$\varepsilon$-伪谱内的特征值周围圆盘的一些新结果。此外，我们研究了满足以下条件的点$z \in \mathbb{C}$构成的集合：对应于预解式最大奇异值的左、右奇异向量的内积的绝对值小于$\varepsilon$。我们通过数值实验证明，该集合的规模几乎可以与“预解式和最近秩一矩阵的相对差小于$\varepsilon$”的集合一样大，并对此给出了部分解释。文中还讨论了一些可能的应用。
英文摘要：
For a square matrix $A$, the resolvent of $A$ at a point $z \in \mathbb{C}$ is defined as $(A-zI )^{-1}$. We consider the set of points $z \in \mathbb{C}$ where the relative difference in 2-norm between the resolvent and the nearest rank one matrix is less than a given number $ε\in (0,1)$. We establish a relationship between this set and the $ε$-pseudospectrum of $A$, and we derive specific results about this set for Jordan blocks and for a class of large Toeplitz matrices. We also derive disks about the eigenvalues of $A$ that are contained in this set, and this leads to some new results on disks about the eigenvalues that are contained in the $ε$-pseudospectrum of $A$. In addition, we consider the set of points $z \in \mathbb{C}$ where the absolute value of the inner product of the left and right singular vectors corresponding to the largest singular value of the resolvent is less than $ε$. We demonstrate numerically that this set can be almost as large as the one where the relative difference between the resolvent and the nearest rank one matrix is less than $ε$ and we give a partial explanation for this. Some possible applications are discussed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
