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Some bounds for the spectral radius of nonnegative tensors

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Abstract

In this paper, we extend the well-known column sum bound of the spectral radius for nonnegative matrices to the tensor case, an upper bound of the spectral radius for a nonnegative tensor is given via the largest eigenvalue of a symmetric tensor. Also we show some bounds of spectral radius of nonnegative tensors based on the sum of the entries in the other indices of tensors. We demonstrate that our new results improve existing results. The other main results of this paper is to provide a sharper Ky Fan type theorem and a comparison theorem for nonnegative tensors. Finally, we make use of our bounds to give a perturbation bound for the spectral radius of symmetric nonnegative tensors. This result is similar to the Weyl theorem for the matrix case.

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Acknowledgments

We are thankful to referees for their valuable suggestions. Also we would like to thank Prof. Liqun Qi for sending us his recent paper [19] and some suggestions, and Prof. Chi-Kwong Li for the discussion when he visited Hong Kong Baptist University.

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, People’s Republic of China

    Wen Li

  2. Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong

    Michael K. Ng

  3. Hong Kong Baptist University Institute of Research and Continuing Education, Shenzhen, China

    Michael K. Ng

Authors
  1. Wen Li
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  2. Michael K. Ng
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Corresponding author

Correspondence to Michael K. Ng.

Additional information

Wen Li was supported in part by National Natural Science Foundations of China (Nos.10971075,11271144), Guangdong Provincial Natural Science Foundations (Nos. s2012010009985, s2013010012530), Research Fund for the Doctoral Program of Higher Education of China (No. 20104407110001) and Project of Department of Education of Guangdong Province (No. 2013KJCX0053)

Michael K. Ng Research supported in part by RGC GRF Grant No. 201812.

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Li, W., Ng, M.K. Some bounds for the spectral radius of nonnegative tensors. Numer. Math. 130, 315–335 (2015). https://doi.org/10.1007/s00211-014-0666-5

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  • DOI: https://doi.org/10.1007/s00211-014-0666-5

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