A note on numerical ranges of tensors
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866916885173370880 |
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| author | Rout, Nirmal Chandra Panigrahy, Krushnachandra Mishra, Debasisha |
| author_facet | Rout, Nirmal Chandra Panigrahy, Krushnachandra Mishra, Debasisha |
| contents | Theory of numerical range and numerical radius for tensors is not studied much in the literature. In 2016, Ke {\it et al.} [Linear Algebra Appl., 508 (2016) 100-132] introduced first the notion of numerical range of a tensor via the $k$-mode product. However, the convexity of the numerical range via the $k$-mode product was not proved by them. In this paper, the notion of numerical range and numerical radius for even-order square tensors using inner product via the Einstein product are introduced first. We provide some sufficient conditions using numerical radius for a tensor to being unitary. The convexity of the numerical range is also proved. We also provide an algorithm to plot the numerical range of a tensor. Furthermore, some properties of the numerical range for the Moore--Penrose inverse of a tensor are discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_08154 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A note on numerical ranges of tensors Rout, Nirmal Chandra Panigrahy, Krushnachandra Mishra, Debasisha Rings and Algebras Spectral Theory 15A69, 15A60 Theory of numerical range and numerical radius for tensors is not studied much in the literature. In 2016, Ke {\it et al.} [Linear Algebra Appl., 508 (2016) 100-132] introduced first the notion of numerical range of a tensor via the $k$-mode product. However, the convexity of the numerical range via the $k$-mode product was not proved by them. In this paper, the notion of numerical range and numerical radius for even-order square tensors using inner product via the Einstein product are introduced first. We provide some sufficient conditions using numerical radius for a tensor to being unitary. The convexity of the numerical range is also proved. We also provide an algorithm to plot the numerical range of a tensor. Furthermore, some properties of the numerical range for the Moore--Penrose inverse of a tensor are discussed. |
| title | A note on numerical ranges of tensors |
| topic | Rings and Algebras Spectral Theory 15A69, 15A60 |
| url | https://arxiv.org/abs/2108.08154 |