A Note on Eigenvalues of Perturbed Hermitian Matrices
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911101681139712 |
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| author | Li, Chi-Kwong Li, Ren-Cang |
| author_facet | Li, Chi-Kwong Li, Ren-Cang |
| contents | Let $$ A=\left(\begin{array}{cc} H_1 & E^*\\ E & H_2\end{array}\right) \quad \hbox{ and } \quad \wtd A=\left(\begin{array}{cc} H_1 & O\\ O & H_2\end{array}\right)$$ be two $N$-by-$N$ Hermitian matrices with eigenvalues $λ_1 \ge \cdots \ge λ_{N}$ and $\wtd λ_1 \ge \cdots \ge \wtd λ_N$, respectively. \iffalse There are two kinds of perturbation bounds on $|λ_i - \wtd λ_i|$:
$|λ_i- \wtd λ_i| \le \|E\|$, where $\|E\|$
is the largest singular value of $\|E\|$, regardless of
$H_i$'s spectral distributions, and
$|λ_i - \wtd λ_i| \le \|E\|^2/η$, where $η$ is
the minimum gap between $H_i$'s spectra. \end{enumerate} Bounds of the first kind overestimate the changes when $\|E\|\llη$ while those of the second kind may blow up when $η$ is too tiny. \fi Denote by $\|E\|$ the spectral norm of the matrix $E$, and $η$ the spectral gap between the spectra of $H_1$ and $H_2$. It is shown that $$ |λ_i - \wtd λ_i| \le {2\|E\|^2 \over η+\sqrt{η^2+4\|E\|^2}} \, , $$ which improves all the existing results. Similar bounds are obtained for singular values of matrices under block perturbations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08203 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Note on Eigenvalues of Perturbed Hermitian Matrices Li, Chi-Kwong Li, Ren-Cang Numerical Analysis 15A42, 15A18, 65F15 Let $$ A=\left(\begin{array}{cc} H_1 & E^*\\ E & H_2\end{array}\right) \quad \hbox{ and } \quad \wtd A=\left(\begin{array}{cc} H_1 & O\\ O & H_2\end{array}\right)$$ be two $N$-by-$N$ Hermitian matrices with eigenvalues $λ_1 \ge \cdots \ge λ_{N}$ and $\wtd λ_1 \ge \cdots \ge \wtd λ_N$, respectively. \iffalse There are two kinds of perturbation bounds on $|λ_i - \wtd λ_i|$: $|λ_i- \wtd λ_i| \le \|E\|$, where $\|E\|$ is the largest singular value of $\|E\|$, regardless of $H_i$'s spectral distributions, and $|λ_i - \wtd λ_i| \le \|E\|^2/η$, where $η$ is the minimum gap between $H_i$'s spectra. \end{enumerate} Bounds of the first kind overestimate the changes when $\|E\|\llη$ while those of the second kind may blow up when $η$ is too tiny. \fi Denote by $\|E\|$ the spectral norm of the matrix $E$, and $η$ the spectral gap between the spectra of $H_1$ and $H_2$. It is shown that $$ |λ_i - \wtd λ_i| \le {2\|E\|^2 \over η+\sqrt{η^2+4\|E\|^2}} \, , $$ which improves all the existing results. Similar bounds are obtained for singular values of matrices under block perturbations. |
| title | A Note on Eigenvalues of Perturbed Hermitian Matrices |
| topic | Numerical Analysis 15A42, 15A18, 65F15 |
| url | https://arxiv.org/abs/2508.08203 |