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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.12408 |
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| _version_ | 1866915742167859200 |
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| author | Temjensangba Mishra, Hemant K. Paul, Niloy |
| author_facet | Temjensangba Mishra, Hemant K. Paul, Niloy |
| contents | Given $2n \times 2n$ real symmetric positive semidefinite matrix $A$ with symplectic kernel, there exists a real $2n \times 2n$ \emph{symplectic matrix} $M$ such that $M^TAM= D \oplus D$, where $D$ is an $n \times n$ non-negative diagonal matrix which is unique up to permutation of its diagonal entries.
The diagonal entries of $D$ are called the \emph{symplectic eigenvalues} or symplectic spectrum of $A$.
In this work, we investigate some majorization and weak supermajorization relations between the symplectic spectra of two positive semidefinite matrices.
More explicitly, suppose $A$ and $B$ are $2n \times 2n$ real symmetric positive semidefinite matrices with symplectic kernels.
We show that if the symplectic spectrum of $A$ is majorized by the symplectic spectrum of $B$, then $A$ lies in the convex hull of the symplectic orbit of $B$.
We also establish that only a weak converse of this statement holds; i.e., if $A$ lies in the convex hull of the symplectic orbit of $B$ then the symplectic spectrum of $A$ is \emph{weakly supermajorized} by the symplectic spectrum of $B$.
Several consequences of our results are also presented.
Our methods make use of well-known connections between the theory of majorization, doubly stochastic, doubly superstochastic, and symplectic matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12408 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Majorization between symplectic spectra of positive semidefinite matrices Temjensangba Mishra, Hemant K. Paul, Niloy Functional Analysis Mathematical Physics Symplectic Geometry Spectral Theory 15B48, 15A18 Given $2n \times 2n$ real symmetric positive semidefinite matrix $A$ with symplectic kernel, there exists a real $2n \times 2n$ \emph{symplectic matrix} $M$ such that $M^TAM= D \oplus D$, where $D$ is an $n \times n$ non-negative diagonal matrix which is unique up to permutation of its diagonal entries. The diagonal entries of $D$ are called the \emph{symplectic eigenvalues} or symplectic spectrum of $A$. In this work, we investigate some majorization and weak supermajorization relations between the symplectic spectra of two positive semidefinite matrices. More explicitly, suppose $A$ and $B$ are $2n \times 2n$ real symmetric positive semidefinite matrices with symplectic kernels. We show that if the symplectic spectrum of $A$ is majorized by the symplectic spectrum of $B$, then $A$ lies in the convex hull of the symplectic orbit of $B$. We also establish that only a weak converse of this statement holds; i.e., if $A$ lies in the convex hull of the symplectic orbit of $B$ then the symplectic spectrum of $A$ is \emph{weakly supermajorized} by the symplectic spectrum of $B$. Several consequences of our results are also presented. Our methods make use of well-known connections between the theory of majorization, doubly stochastic, doubly superstochastic, and symplectic matrices. |
| title | Majorization between symplectic spectra of positive semidefinite matrices |
| topic | Functional Analysis Mathematical Physics Symplectic Geometry Spectral Theory 15B48, 15A18 |
| url | https://arxiv.org/abs/2601.12408 |