\(Z\)-Eigenvalue Localization Sets for Even Order Tensors and Their Applications

Abstract

Firstly, a new Geršgorin-type \(Z\)-eigenvalue localization set with parameters for even order tensors is presented. As an application, some sufficient conditions for the positive (semi-)definiteness of even order real symmetric tensors are obtained. Secondly, by selecting appropriate parameters an optimal set is obtained and proved to be tighter than some existing results. Thirdly, as another application, new upper bounds for the \(Z\)-spectral radius of even order weakly symmetric nonnegative tensors are obtained. Finally, numerical examples are given to verify the theoretical results.

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Acknowledgements

The authors are grateful to the referees and Editors-in-Chief John King, Benoît Perthame for their comments and suggestions. This work is supported by Science and Technology Projects of Education Department of Guizhou Province (Grant No. KY[2015]352); Science and Technology Top-notch Talents Support Project of Education Department of Guizhou Province (Grant No. QJHKYZ [2016]066); National Natural Science Foundations of China (Grant No. 11501141) and Natural Science Foundation of Guizhou Minzu University.

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Correspondence to Zhen Chen.

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Sang, C., Chen, Z. \(Z\)-Eigenvalue Localization Sets for Even Order Tensors and Their Applications. Acta Appl Math 169, 323–339 (2020). https://doi.org/10.1007/s10440-019-00300-1

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Keywords

  • Nonnegative tensors
  • \(Z\)-eigenvalues
  • \(Z\)-spectral radius
  • Localization sets
  • Positive definiteness

Mathematics Subject Classification (2010)

  • 15A18
  • 15A42
  • 15A69