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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2603.27634 |
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| _version_ | 1866915897451479040 |
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| author | Temjensangba Mishra, Hemant Kumar |
| author_facet | Temjensangba Mishra, Hemant Kumar |
| contents | Let $A = \begin{bmatrix} E & F \\ F^T & G \end{bmatrix}$ be a $2n \times 2n$ real positive definite matrix, where $E, F,$ and $G$ are $n \times n$ blocks. It is shown that
$\ d(E \oplus G) \prec^w d(A)$.
Here $d(A)$ denotes the $n$-vector consisting of the symplectic eigenvalues of $A$ arranged in the non-decreasing order. We also observe the following weak supermajorization relation, which is interesting on its own:
$ λ\left( \left(\mathscr{C}(G)^{1/2} \mathscr{C}(E) \mathscr{C}(G)^{1/2}\right)^{1/2} \right) \prec^w λ\left( \left(G^{1/2} E G^{1/2} \right)^{1/2} \right)$.
Here $λ\left( \left( G^{1/2}E G^{1/2} \right)^{1/2} \right)$ denotes the $n$-vector with entries given by the eigenvalues of $\left( G^{1/2}E G^{1/2} \right)^{1/2}$ in the non-decreasing order. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27634 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weak supermajorization between symplectic spectra of positive definite matrix and its pinching Temjensangba Mishra, Hemant Kumar Functional Analysis Mathematical Physics Symplectic Geometry Spectral Theory 15B48, 15A18, 15A42 Let $A = \begin{bmatrix} E & F \\ F^T & G \end{bmatrix}$ be a $2n \times 2n$ real positive definite matrix, where $E, F,$ and $G$ are $n \times n$ blocks. It is shown that $\ d(E \oplus G) \prec^w d(A)$. Here $d(A)$ denotes the $n$-vector consisting of the symplectic eigenvalues of $A$ arranged in the non-decreasing order. We also observe the following weak supermajorization relation, which is interesting on its own: $ λ\left( \left(\mathscr{C}(G)^{1/2} \mathscr{C}(E) \mathscr{C}(G)^{1/2}\right)^{1/2} \right) \prec^w λ\left( \left(G^{1/2} E G^{1/2} \right)^{1/2} \right)$. Here $λ\left( \left( G^{1/2}E G^{1/2} \right)^{1/2} \right)$ denotes the $n$-vector with entries given by the eigenvalues of $\left( G^{1/2}E G^{1/2} \right)^{1/2}$ in the non-decreasing order. |
| title | Weak supermajorization between symplectic spectra of positive definite matrix and its pinching |
| topic | Functional Analysis Mathematical Physics Symplectic Geometry Spectral Theory 15B48, 15A18, 15A42 |
| url | https://arxiv.org/abs/2603.27634 |