Abstract
The tensor eigenvalue problem has been widely studied in recent years. In this paper, we propose a new constructive way to estimate bounds on the H-eigenvalues of two kinds of interval tensors including nonnegative tensors and real symmetric tensors of even order. As applications, perturbation analysis of both nonnegative tensors and real symmetric tensors is numerically studied. One open question in Qi (Linear Algebra Appl 439:228–238, 2013) has also been answered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability statement
All data generated or analyzed during this study are included in this article.
References
Bozorgmanesh H, Hajarian M, Chronopoulos A (2020) Interval tensors and their application in solving multi-linear systems of equations. Comput Math Appl 79:697–715
Chang K, Pearson K, Zhang T (2008) Perron–Frobenius theorem for nonnegative tensors. Commun Math Sci 6:507–520
Chang K, Qi L, Zhou G (2010) Singular value of a real rectangular tensor. J Math Anal Appl 370:284–294
Cox D, Little J, O’Shea D (1998) Using algebraic geometry. Springer, Berlin
Cui L, Song Y (2019) On the uniqueness of the positive Z-eigenvector for nonnegative tensors. J Comput Appl Math 352:72–78
Cui L, Chen C, Li W, Ng M (2016) An eigenvalue problem for even order tensors with its applications. Linear Multilinear A 64:602–621
Cui L, Li M, Song Y (2019) Preconditioned tensor splitting iterations method for solving multi-linear systems. Appl Math Lett 96:89–94
Dahl G, Leinass JM, Myrheim J, Ovrum E (2007) A tensor product matrix approximation problem in quantum physics. Linear Algebra Appl 420:711–725
De Lathauwer L, De Moor B, Vandewalle J (2000) A multilinear singular value decomposition. SIAM J Matrix Anal Appl 21:1253–1278
Drineas P, Lim L (2005) A multilinear spectral theory of hypergraphs and expander hypergraphs, preprint. Stanford University, Stanford
Friedland S, Gaubert S, Han L (2009) Perron–Frobenius theorem for nonnegative multilinear forms and extensions. Linear Algebra Appl 438:738–749
Horn R, Johnson C (1985) Matrix analysis. Cambridge University Press, Cambridge
Jiang Z, Li J (2021) Solving tensor absolute value equation. Appl Numer Math 170:255–268
Kofidis E, Regalia Ph (2002) On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J Matrix Anal Appl 23:863–884
Li W, Ng M (2015) Some bounds for the spectral radius of nonnegative tensors. Numer Math 130:315–335
Li C, Li Y, Kong X (2014) New eigenvalue inclusion sets for tensors. Numer Linear Algebra Appl 21:39–50
Li C, Chen Z, Li Y (2015) A new eigenvalue inclusion set for tensors and its applications. Linear Algebra Appl 481:36–53
Lim L (2005) Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE international workshop on computational advances in multi-sensor adaptive processing (CAMSAP ’05), Puerto Vallarta, Mexico, pp 129–132. https://doi.org/10.1109/CAMAP.2005.1574201
Ng M, Qi L, Zhou G (2009) Finding the largest eigenvalue of a nonnegative tensor. SIAM J Matrix Anal Appl 31:1090–1099
Pearson K (2010) Primitive tensors and convergence of an iterative process for the eigenvalues of a primitive tensor. arXiv:1004.2423
Qi L (2005) Eigenvalues of a real supersymmetric tensor. J Symb Comput 40:1302–1324
Qi L (2007) Eigenvalues and invariants of tensors. J Math Anal Appl 325:1363–1377
Qi L (2013) Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl 439:228–238
Yang Y, Yang Q (2010) Further results for Perron–Frobenius theorem for nonnegative tensors. SIAM J Matrix Anal Appl 31:2517–2530
Yang J, Zhao X, Mei J, Wang S, Ma T, Huang T (2019) Total variation and high-order total variation adaptive model for restoring blurred images with Cauchy noise. Comput Math Appl 77:1255–1272
Yang J, Zhao X, Ma T, Chen Y, Huang T, Ding M (2020) Remote sensing image destriping using unidirectional high-order total variation and nonconvex low-rank regularization. J Comput Appl Math 363:124–144
Zhao J (2020) E-eigenvalue localization sets for fourth-order tensors. Bull Malays Math Sci Soc 43:1685–1707. https://doi.org/10.1007/s40840-019-00768-y
Acknowledgements
The author would like to thank the editor and anonymous referees for their valuable comments and suggestions, which greatly improve the quality of the original manuscript.
Funding
This research is supported in part by Foundation of Henan Educational Committee (no. 21A110013), Foundation of Henan Normal University (no. 2021PL03), and Xinxiang Aviation Industry (Group) Co., Ltd. (no. 2023003).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have not disclosed any competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Cui, LB., Zhang, XJ. Bounds of H-eigenvalues of interval tensors. Comp. Appl. Math. 42, 280 (2023). https://doi.org/10.1007/s40314-023-02418-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-023-02418-3