When is the Resolvent Like a Rank One Matrix?
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| Format: | Preprint |
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2025
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| author | Greenbaum, Anne Kyanfar, Faranges Salemi, Abbas |
| author_facet | Greenbaum, Anne Kyanfar, Faranges Salemi, Abbas |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_07686 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | When is the Resolvent Like a Rank One Matrix? Greenbaum, Anne Kyanfar, Faranges Salemi, Abbas Numerical Analysis 65F99 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. |
| title | When is the Resolvent Like a Rank One Matrix? |
| topic | Numerical Analysis 65F99 |
| url | https://arxiv.org/abs/2501.07686 |