Mathematics (May 2025)

T-Eigenvalues of Third-Order Quaternion Tensors

  • Zhuo-Heng He,
  • Mei-Ling Deng,
  • Shao-Wen Yu

DOI
https://doi.org/10.3390/math13101549
Journal volume & issue
Vol. 13, no. 10
p. 1549

Abstract

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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.

Keywords