On the Continuity of Schur-Horn Mapping
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908745056911360 |
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| author | Chen, Hengzhun Li, Yingzhou |
| author_facet | Chen, Hengzhun Li, Yingzhou |
| contents | The Schur-Horn theorem is a well-known result that characterizes the relationship between the diagonal elements and eigenvalues of a symmetric (Hermitian) matrix. In this paper, we extend this theorem by exploring the eigenvalue perturbation of a symmetric (Hermitian) matrix with fixed diagonals, which is referred to as the continuity of the Schur-Horn mapping. We introduce a concept called strong Schur-Horn continuity, characterized by minimal constraints on the perturbation. We demonstrate that several categories of matrices exhibit strong Schur-Horn continuity. Leveraging this notion, along with a majorization constraint on the perturbation, we prove the Schur-Horn continuity for general symmetric (Hermitian) matrices. The Schur-Horn continuity finds applications in oblique manifold optimization related to quantum computing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00701 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Continuity of Schur-Horn Mapping Chen, Hengzhun Li, Yingzhou Numerical Analysis Quantum Physics The Schur-Horn theorem is a well-known result that characterizes the relationship between the diagonal elements and eigenvalues of a symmetric (Hermitian) matrix. In this paper, we extend this theorem by exploring the eigenvalue perturbation of a symmetric (Hermitian) matrix with fixed diagonals, which is referred to as the continuity of the Schur-Horn mapping. We introduce a concept called strong Schur-Horn continuity, characterized by minimal constraints on the perturbation. We demonstrate that several categories of matrices exhibit strong Schur-Horn continuity. Leveraging this notion, along with a majorization constraint on the perturbation, we prove the Schur-Horn continuity for general symmetric (Hermitian) matrices. The Schur-Horn continuity finds applications in oblique manifold optimization related to quantum computing. |
| title | On the Continuity of Schur-Horn Mapping |
| topic | Numerical Analysis Quantum Physics |
| url | https://arxiv.org/abs/2407.00701 |