A new characterization of symmetric $H^+$-tensors and $M$-tensors
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866915353674645504 |
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| author | Shi, Xin Zuluaga, Luis F. |
| author_facet | Shi, Xin Zuluaga, Luis F. |
| contents | In this work, we present a new characterization of symmetric $H^+$-tensors. It is known that a symmetric tensor is an $H^+$-tensor if and only if it is a generalized diagonally dominant tensor with nonnegative diagonal elements. By exploring the diagonal dominance property, we derive new necessary and sufficient conditions for a symmetric tensor to be an $H^+$-tensor. Based on these conditions, we propose a novel method that allows to check if a tensor is a symmetric $H^+$-tensor in polynomial time. Moreover, these results can be applied to the closely related and important class of $M$-tensors. In particular, this allows to efficiently compute the minimum $H$-eigenvalue of symmetric $M$-tensors. Furthermore, we show how this latter result can be used to provide tighter lower bounds for the minimum $H$-eigenvalue of the Fan product of two symmetric $M$-tensors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_08206 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A new characterization of symmetric $H^+$-tensors and $M$-tensors Shi, Xin Zuluaga, Luis F. Spectral Theory Optimization and Control 15A69 In this work, we present a new characterization of symmetric $H^+$-tensors. It is known that a symmetric tensor is an $H^+$-tensor if and only if it is a generalized diagonally dominant tensor with nonnegative diagonal elements. By exploring the diagonal dominance property, we derive new necessary and sufficient conditions for a symmetric tensor to be an $H^+$-tensor. Based on these conditions, we propose a novel method that allows to check if a tensor is a symmetric $H^+$-tensor in polynomial time. Moreover, these results can be applied to the closely related and important class of $M$-tensors. In particular, this allows to efficiently compute the minimum $H$-eigenvalue of symmetric $M$-tensors. Furthermore, we show how this latter result can be used to provide tighter lower bounds for the minimum $H$-eigenvalue of the Fan product of two symmetric $M$-tensors. |
| title | A new characterization of symmetric $H^+$-tensors and $M$-tensors |
| topic | Spectral Theory Optimization and Control 15A69 |
| url | https://arxiv.org/abs/2008.08206 |