Positive Definiteness of $4$th Order $3$-Dimensional Symmetric Tensors with entries $-1$, $0$, $1$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917946323894272 |
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| author | Ye, Li Song, Yisheng |
| author_facet | Ye, Li Song, Yisheng |
| contents | It is well-known that a symmetric matrix with its entries $\pm1$ is not positive definite. But this is not ture for symmetric tensors (hyper-matrix). In this paper, we mainly dicuss the positive (semi-)definiteness criterion of a class of $4$th order $3$-dimensional symmetric tensors with entries $t_{ijkl}\in\{-1,0,1\}$. Through theoretical derivations and detailed classification discussions, the criterion for determining the positive (semi-)definiteness of such a class of tensors are provided based on the relationships and number values of its entries. Which establishes some unique properties of higher symmetric tensors that distinct from ones of matrces |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_03127 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Positive Definiteness of $4$th Order $3$-Dimensional Symmetric Tensors with entries $-1$, $0$, $1$ Ye, Li Song, Yisheng Optimization and Control Algebraic Geometry 15A69, 90C23, 15A72, 15A63, 90C20, 90C30 It is well-known that a symmetric matrix with its entries $\pm1$ is not positive definite. But this is not ture for symmetric tensors (hyper-matrix). In this paper, we mainly dicuss the positive (semi-)definiteness criterion of a class of $4$th order $3$-dimensional symmetric tensors with entries $t_{ijkl}\in\{-1,0,1\}$. Through theoretical derivations and detailed classification discussions, the criterion for determining the positive (semi-)definiteness of such a class of tensors are provided based on the relationships and number values of its entries. Which establishes some unique properties of higher symmetric tensors that distinct from ones of matrces |
| title | Positive Definiteness of $4$th Order $3$-Dimensional Symmetric Tensors with entries $-1$, $0$, $1$ |
| topic | Optimization and Control Algebraic Geometry 15A69, 90C23, 15A72, 15A63, 90C20, 90C30 |
| url | https://arxiv.org/abs/2503.03127 |