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T-Eigenvalues of Third-Order Quaternion Tensors

Author

Listed:
  • Zhuo-Heng He

    (Department of Mathematics and Newtouch Center for Mathematics, Shanghai University, Shanghai 200444, China
    Sino-European School of Technology, Shanghai University, Shanghai 200444, China)

  • Mei-Ling Deng

    (Department of Mathematics and Newtouch Center for Mathematics, Shanghai University, Shanghai 200444, China)

  • Shao-Wen Yu

    (School of Mathematics, East China University of Science and Technology, Shanghai 200237, China)

Abstract

In this paper, theories, algorithms and properties of eigenvalues of quaternion tensors via the t-product termed T-eigenvalues are explored. Firstly, we define the T-eigenvalue of quaternion tensors and provide an algorithm to compute the right T-eigenvalues and the corresponding T-eigentensors, along with an example to illustrate the efficiency of our algorithm by comparing it with other methods. We then study some inequalities related to the right T-eigenvalues of Hermitian quaternion tensors, providing upper and lower bounds for the right T-eigenvalues of the sum of a pair of Hermitian tensors. We further generalize the Weyl theorem from matrices to quaternion third-order tensors. Additionally, we explore estimations related to right T-eigenvalues, extending the Geršgorin theorem for matrices to quaternion third-order tensors.

Suggested Citation

  • Zhuo-Heng He & Mei-Ling Deng & Shao-Wen Yu, 2025. "T-Eigenvalues of Third-Order Quaternion Tensors," Mathematics, MDPI, vol. 13(10), pages 1-17, May.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:10:p:1549-:d:1651698
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    References listed on IDEAS

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    1. Fengsheng Wu & Chaoqian Li & Yaotang Li & Niansheng Tang, 2025. "Complex Representation Matrix of Third-Order Quaternion Tensors with Application to Video Inpainting," Journal of Optimization Theory and Applications, Springer, vol. 205(2), pages 1-33, May.
    2. Hua Zhou & Lexin Li & Hongtu Zhu, 2013. "Tensor Regression with Applications in Neuroimaging Data Analysis," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(502), pages 540-552, June.
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