Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916283750023168 |
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| author | Tarnowski, Wojciech |
| author_facet | Tarnowski, Wojciech |
| contents | Sensitivity of an eigenvalue $λ_i$ to the perturbation of matrix elements is controlled by the eigenvalue condition number defined as $κ_i = \sqrt{\left< L_i | L_i\right> \left< R_i|R_i \right> }$, where $\left<L_i\right|$ and $\left|R_i\right>$ are left and right eigenvectors to the eigenvalue $λ_i$. In Random Matrix Theory the squared eigenvalue condition number is also known as the eigenvector self-overlap. In this work we calculate the asymptotics of the joint probability density function of the real eigenvalue and the square of the corresponding eigenvalue condition number for the real elliptic ensemble in the double scaling regime of almost Hermiticity and close to the edge of the spectrum. As a byproduct, we also calculate the one-parameter deformation of the Scorer's function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_03249 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge Tarnowski, Wojciech Mathematical Physics Sensitivity of an eigenvalue $λ_i$ to the perturbation of matrix elements is controlled by the eigenvalue condition number defined as $κ_i = \sqrt{\left< L_i | L_i\right> \left< R_i|R_i \right> }$, where $\left<L_i\right|$ and $\left|R_i\right>$ are left and right eigenvectors to the eigenvalue $λ_i$. In Random Matrix Theory the squared eigenvalue condition number is also known as the eigenvector self-overlap. In this work we calculate the asymptotics of the joint probability density function of the real eigenvalue and the square of the corresponding eigenvalue condition number for the real elliptic ensemble in the double scaling regime of almost Hermiticity and close to the edge of the spectrum. As a byproduct, we also calculate the one-parameter deformation of the Scorer's function. |
| title | Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2401.03249 |