Completely Positive Biquadratic Tensors
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917007352397824 |
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| author | Qi, Liqun Cui, Chunfeng Chen, Haibin Xu, Yi |
| author_facet | Qi, Liqun Cui, Chunfeng Chen, Haibin Xu, Yi |
| contents | In this paper, we systemically introduce completely positive biquadratic (CPB) tensors and copositive biquadratic tensors. We show that all weakly CPB tensors are sum of squares tensors, the CPB tensor cone and the copositive biquadratic tensor cone are dual cone to each other. We also show that the outer product of two completely positive matrices is a CPB tensor, and the outer product of two copositive matrices is a copositive biquadratic tensor. We then study two easily checkable subclasses of CPB tensors, namely positive biquadratic Cauchy tensors and biquadratic Pascal tensors. We show that a biquadratic Pascal tensor is both strongly CPB and positive definite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10972 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Completely Positive Biquadratic Tensors Qi, Liqun Cui, Chunfeng Chen, Haibin Xu, Yi Rings and Algebras In this paper, we systemically introduce completely positive biquadratic (CPB) tensors and copositive biquadratic tensors. We show that all weakly CPB tensors are sum of squares tensors, the CPB tensor cone and the copositive biquadratic tensor cone are dual cone to each other. We also show that the outer product of two completely positive matrices is a CPB tensor, and the outer product of two copositive matrices is a copositive biquadratic tensor. We then study two easily checkable subclasses of CPB tensors, namely positive biquadratic Cauchy tensors and biquadratic Pascal tensors. We show that a biquadratic Pascal tensor is both strongly CPB and positive definite. |
| title | Completely Positive Biquadratic Tensors |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2510.10972 |