The spectral map for weighted Cauchy matrices is an involution
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915330150891520 |
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| author | Pushnitski, Alexander Treil, Sergei |
| author_facet | Pushnitski, Alexander Treil, Sergei |
| contents | Let $N$ be a natural number. We consider weighted Cauchy matrices of the form \[ \mathcal{C}_{a,A}=\left\{\frac{\sqrt{A_j A_k}}{a_k+a_j}\right\}_{j,k=1}^N, \] where $A_1,\dots,A_N$ are positive real numbers and $a_1,\dots,a_N$ are distinct positive real numbers, listed in increasing order. Let $b_1,\dots,b_N$ be the eigenvalues of $\mathcal{C}_{a,A}$, listed in increasing order. Let $B_k$ be positive real numbers such that $\sqrt{B_k}$ is the Euclidean norm of the orthogonal projection of the vector \[ v_A=(\sqrt{A_1},\dots,\sqrt{A_N}) \] onto the $k$'th eigenspace of $\mathcal{C}_{a,A}$. We prove that the spectral map $(a,A)\mapsto (b,B)$ is an involution and discuss simple properties of this map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18707 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The spectral map for weighted Cauchy matrices is an involution Pushnitski, Alexander Treil, Sergei Rings and Algebras Numerical Analysis 15B05 Let $N$ be a natural number. We consider weighted Cauchy matrices of the form \[ \mathcal{C}_{a,A}=\left\{\frac{\sqrt{A_j A_k}}{a_k+a_j}\right\}_{j,k=1}^N, \] where $A_1,\dots,A_N$ are positive real numbers and $a_1,\dots,a_N$ are distinct positive real numbers, listed in increasing order. Let $b_1,\dots,b_N$ be the eigenvalues of $\mathcal{C}_{a,A}$, listed in increasing order. Let $B_k$ be positive real numbers such that $\sqrt{B_k}$ is the Euclidean norm of the orthogonal projection of the vector \[ v_A=(\sqrt{A_1},\dots,\sqrt{A_N}) \] onto the $k$'th eigenspace of $\mathcal{C}_{a,A}$. We prove that the spectral map $(a,A)\mapsto (b,B)$ is an involution and discuss simple properties of this map. |
| title | The spectral map for weighted Cauchy matrices is an involution |
| topic | Rings and Algebras Numerical Analysis 15B05 |
| url | https://arxiv.org/abs/2504.18707 |