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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2412.17368 |
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| _version_ | 1866917876782333952 |
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| author | Qi, Liqun Cui, Chunfeng Luo, Ziyan |
| author_facet | Qi, Liqun Cui, Chunfeng Luo, Ziyan |
| contents | We define lower triangular tensors, and show that all diagonal entries of such a tensor are eigenvalues of that tensor. We then define lower triangular sub-symmetric tensors, and show that the number of independent entries of a lower triangular sub-symmetric tensor is the same as that of a symmetric tensor of the same order and dimension. We further introduce third order Hermitian tensors, third order positive semi-definite Hermitian tensors, and third order positive semi-definite symmetric tensors. Third order completely positive tensors are positive semi-definite symmetric tensors. Then we show that a third order positive semi-definite Hermitian tensor is triangularly decomposable. This generalizes the classical result of Cholesky decomposition in matrix analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17368 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Triangular Decomposition of Third Order Hermitian Tensors Qi, Liqun Cui, Chunfeng Luo, Ziyan Rings and Algebras We define lower triangular tensors, and show that all diagonal entries of such a tensor are eigenvalues of that tensor. We then define lower triangular sub-symmetric tensors, and show that the number of independent entries of a lower triangular sub-symmetric tensor is the same as that of a symmetric tensor of the same order and dimension. We further introduce third order Hermitian tensors, third order positive semi-definite Hermitian tensors, and third order positive semi-definite symmetric tensors. Third order completely positive tensors are positive semi-definite symmetric tensors. Then we show that a third order positive semi-definite Hermitian tensor is triangularly decomposable. This generalizes the classical result of Cholesky decomposition in matrix analysis. |
| title | Triangular Decomposition of Third Order Hermitian Tensors |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2412.17368 |