Some Bounds of Eigenvalues for Hadamard Product and Fan Product of Tensors

Abstract

In this paper, some new upper bounds on the spectral radius of Hadamard product of nonnegative tensors are given. To show their sharpness, the comparisons among these bounds, including the existing one by Sun et al. (Linear Multilinear Algebra 66:1199–1214, 2018), are performed. We also present some lower bounds on the minimum eigenvalue of Fan product of irreducible strong \({{\mathcal {M}}}\)-tensors and their sharpness under different conditions are investigated. Some numerical examples are provided to illustrate our theoretical results.

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Acknowledgements

The authors would like to thank the anonymous referees who made much useful and detailed suggestions that helped us to improve the quality of this paper. This work was supported by the National Natural Science Foundation of China (No. 11571004) and the Fundamental Research Funds for the Central Universities (lzujbky-2017-it54).

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Correspondence to Bing Zheng.

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Communicated by Ali Armandnejad.

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Xu, Y., Zheng, B. & Zhao, R. Some Bounds of Eigenvalues for Hadamard Product and Fan Product of Tensors. Bull. Iran. Math. Soc. 46, 1003–1026 (2020). https://doi.org/10.1007/s41980-019-00307-2

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Keywords

  • Hadamard product
  • Fan product
  • Nonnegative tensor
  • Strong \({{\mathcal {M}}}\)-tensor
  • Eigenvalue
  • Upper and lower bounds

Mathematics Subject Classification

  • 15A69
  • 15A18
  • 15A42