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Research Article

On C-tensor and its application to eigenvalue localization

ORCID Icon, ORCID Icon &
Pages 6279-6296 | Received 15 Aug 2020, Accepted 03 Jun 2021, Published online: 12 Jul 2021
 

Abstract

The notions of C-tensor, C0-tensor and C¯-tensor are introduced first. Different necessary and sufficient conditions for a tensor to be a C-tensor, C0-tensor and C¯-tensor are provided. We next show that the sum of two C-tensors (C0-tensors) is a C-tensor (C0-tensor) while the Hadamard product of two C-tensors (C0-tensors) is not a C-tensor (C0-tensor). We also present a result that illustrates the Hadamard product of two C-tensor is again a C-tensor under some sufficient conditions. As an application of these classes of tensors, an exclusion interval for the real eigenvalues of a real tensor is proposed. Finally, we provide a necessary and sufficient condition for the exclusion interval to be nonempty.

2010 Mathematics Subject Classifications:

Acknowledgments

The authors would like to thank the anonymous referee for the valuable comments and suggestions. The third author has been supported by the Spanish research Grant PGC2018-096321-B-I00 (MCIU/AEI), by Gobierno de Aragón (E41-17R) and Feder 2014-2020. The first two authors have been supported by the grant CRG/2018/002986 from the Science and Engineering Research Board, India. The first two authors would also like to thank the Government of India for introducing the work from home initiative during the COVID-19 pandemic.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The third author has been supported by the Spanish research Grant PGC2018-096321-B-I00 (MCIU/AEI), by Gobierno de Aragón (E41-17R) and Feder 2014-2020. The first two authors have been supported by the grant CRG/2018/002986 from the Science and Engineering Research Board, India.

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