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An Inequality for the Perron Pair of an Irreducible and Symmetric Nonnegative Tensor with Application

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Abstract

The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingular \(\mathscr {M}\)-tensor. We analyze the analytical property of an algebraic simple eigenvalue of symmetric tensors. We also derive an inequality about the Perron pair of nonnegative tensors based on plane stochastic tensors. We finally consider the perturbation of the smallest eigenvalue of nonsingular \(\mathscr {M}\)-tensors and design a strategy to compute its smallest eigenvalue. We verify our results via random numerical examples.

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Acknowledgments

The authors would like to thank Professors L. Qi, M. K. Ng, and Q. Yang for their useful comments on our manuscript, and to thank the anonymous referees for their valuable suggestions which help us to improve the manuscript.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Mao-Lin Che

  2. School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, China

    Yi-Min Wei

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  1. Mao-Lin Che
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  2. Yi-Min Wei
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Correspondence to Yi-Min Wei.

Additional information

The authors were supported by the National Natural Science Foundation of China (No. 11271084) and International Cooperation Project of Shanghai Municipal Science and Technology Commission (No. 16510711200).

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Che, ML., Wei, YM. An Inequality for the Perron Pair of an Irreducible and Symmetric Nonnegative Tensor with Application. J. Oper. Res. Soc. China 5, 65–82 (2017). https://doi.org/10.1007/s40305-016-0138-y

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  • DOI: https://doi.org/10.1007/s40305-016-0138-y

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