Skip to main content
Log in

Eigenvalue bounds of third-order tensors via the minimax eigenvalue of symmetric matrices

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

Upper and lower bounds for H-eigenvalues, Z-spectral radius and C-spectral radius of a third-order tensor are given by the minimax eigenvalue of symmetric matrices extracted from this given tensor. As applications, a sufficient condition for third-order nonsingular M-tensors and some valid sufficient conditions for the uniqueness and solvability of the solutions to multi-linear systems, tensor complementarity problems and non-homogeneous systems are proposed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Benson AR (2019) Three hypergraph eigenvector centralities. SIAM J Math Data Sci 1(2):293–312

    MathSciNet  Google Scholar 

  • Benson AR, Gleich DF (2019) Computing tensor Z-eigenvectors with dynamical systems. SIAM J Matrix Anal Appl 40(4):1311–1324

    MathSciNet  MATH  Google Scholar 

  • Bu C, Wei Y, Sun L et al (2015) Brualdi-type eigenvalue inclusion sets of tensors. Linear Algebra Appl 480:168–175

    MathSciNet  MATH  Google Scholar 

  • Chang J, Chen Y, Qi L (2016) Computing eigenvalues of large scale sparse tensors arising from a hypergraph. SIAM J Sci Comput 38(6):A3618–A3643

    MathSciNet  MATH  Google Scholar 

  • Che H, Chen H, Wang Y (2019) C-eigenvalue inclusion theorems for piezoelectric-type tensors. Appl Math Lett 89:41–49

    MathSciNet  MATH  Google Scholar 

  • Chen Y, Qi L, Virga EG (2018) Octupolar tensors for liquid. J Phys A Math Theor. https://doi.org/10.1088/1751-8121/aa98a8

    MATH  Google Scholar 

  • Chen Y, Jákli A, Qi L (2017) Spectral analysis of piezoelectric tensors. arXiv preprint arXiv: 1703.07937

  • Cooper J, Dutle A (2012) Spectra of uniform hypergraphs. Linear Algebra Appl 436(9):3268–3292

    MathSciNet  MATH  Google Scholar 

  • Cui C, Dai Y, Nie J (2014) All real eigenvalues of symmetric tensors. SIAM J Matrix Anal Appl 35(4):1582–1601

    MathSciNet  MATH  Google Scholar 

  • Curie J, Curie P (1880) Développement par compression de l’électricité polaire dans les cristaux hémièdres à faces inclinées. Bull Minéral 3(4):90–93

    MATH  Google Scholar 

  • De Jong M, Chen W, Geerlings H et al (2015) A database to enable discovery and design of piezoelectric materials. Sci Data 2(1):1–13

    Google Scholar 

  • Ding W, Wei Y (2015) Generalized tensor eigenvalue problems. SIAM J Matrix Anal Appl 36(3):1073–1099

    MathSciNet  MATH  Google Scholar 

  • Ding W, Wei Y (2016) Solving multi-linear systems with M-tensors. J Sci Comput 68(2):689–715

    MathSciNet  MATH  Google Scholar 

  • Grozdanov S, Kaplis N (2016) Constructing higher-order hydrodynamics: the third order. Phys Rev D. https://doi.org/10.1103/PhysRevD.93.066012

    MathSciNet  Google Scholar 

  • He J, Huang T (2014) Upper bound for the largest Z-eigenvalue of positive tensors. Appl Math Lett 38:110–114

    MathSciNet  MATH  Google Scholar 

  • Huang Z, Wang L, Xu Z et al (2018) Some new inequalities for the minimum H-eigenvalue of nonsingular M-tensors. Linear Algebra Appl 558:146–173

    MathSciNet  MATH  Google Scholar 

  • Huang Z, Wang L, Xu Z, Cui J (2019) Some new Z-eigenvalue localization sets for tensors and their applications. Revista de la Unión Matemática Argentina 60(1):99–119

    MathSciNet  MATH  Google Scholar 

  • Kilmer ME, Braman K, Hao N et al (2013) Third-order tensors as operators on matrices: a theoretical and computational framework with applications in imaging. SIAM J Matrix Anal Appl 34(1):148–172

    MathSciNet  MATH  Google Scholar 

  • Kolda TG, Mayo JR (2011) Shifted power method for computing tensor eigenpairs. SIAM J Matrix Anal Appl 32(4):1095–1124

    MathSciNet  MATH  Google Scholar 

  • Li W, Ng MK (2014) On the limiting probability distribution of a transition probability tensor. Linear Multilinear A 62(3):362–385

    MathSciNet  MATH  Google Scholar 

  • Li X, Ng MK (2015) Solving sparse non-negative tensor equations: algorithms and applications. Front Math China 10(3):649–680

    MathSciNet  MATH  Google Scholar 

  • Li C, Li Y, Kong X (2014) New eigenvalue inclusion sets for tensors. Numer Linear Algebra 21(1):39–50

    MathSciNet  MATH  Google Scholar 

  • Li W, Liu D, Vong SW (2015) Z-eigenpair bounds for an irreducible nonnegative tensor. Linear Algebra Appl 483:182–199

    MathSciNet  MATH  Google Scholar 

  • Li C, Liu Y, Li Y (2019) C-eigenvalues intervals for piezoelectric-type tensors. Appl Math Comput 358:244–250

    MathSciNet  MATH  Google Scholar 

  • Li C, Liu Q, Wei Y (2019) Pseudospectra localizations for generalized tensor eigenvalues to seek more positive definite tensors. Comput Appl Math 38:183. https://doi.org/10.1007/s40314-019-0958-6

    MathSciNet  MATH  Google Scholar 

  • Lim LH (2005) Singular values and eigenvalues of tensors: a variational approach. In: 1st IEEE international workshop on computational advances in multi-sensor adaptive processing. IEEE, pp 129–132

  • Luo Z, Qi L, Xiu N (2017) The sparsest solutions to Z-tensor complementarity problems. Optim Lett 11(3):471–482

    MathSciNet  MATH  Google Scholar 

  • Ng M, Qi L, Zhou G (2010) Finding the largest eigenvalue of a nonnegative tensor. SIAM J Matrix Anal Appl 31(3):1090–1099

    MathSciNet  MATH  Google Scholar 

  • Nye JF (1985) Physical properties of crystals: their representation by tensors and matrices. Oxford University Press, Oxford

    MATH  Google Scholar 

  • Padhy S, Dandapat S (2017) Third-order tensor based analysis of multilead ecg for classification of myocardial infarction. Biomed Signal Proces 31:71–78

    Google Scholar 

  • Qi L (2005) Eigenvalues of a real supersymmetric tensor. J Symb Comput 40(6):1302–1324

    MathSciNet  MATH  Google Scholar 

  • Qi L (2007) Eigenvalues and invariants of tensors. J Math Anal Appl 325(2):1363–1377

    MathSciNet  MATH  Google Scholar 

  • Qi L, Teo KL (2003) Multivariate polynomial minimization and its application in signal processing. J Glob Optim 26(4):419–433

    MathSciNet  MATH  Google Scholar 

  • Qi L, Wang Y, Wu EX (2008) D-eigenvalues of diffusion kurtosis tensors. J Comput Appl Math 221(1):150–157

    MathSciNet  MATH  Google Scholar 

  • Qi L, Yu G, Wu EX (2010) Higher order positive semidefinite diffusion tensor imaging. SIAM J Imaging Sci 3(3):416–433

    MathSciNet  MATH  Google Scholar 

  • Qi L, Chen H, Chen Y (2018) Fourth order tensors in physics and mechanics. Tensor eigenvalues and their applications. Springer, Singapore, pp 249–284

    MATH  Google Scholar 

  • Raftery A, Tavaré S (1994) Estimation and modelling repeated patterns in high order markov chains with the mixture transition distribution model. J R Stat Soc C Appl 43(1):179–199

    MATH  Google Scholar 

  • Royer JP, Thirion-Moreau N, Comon P (2011) Computing the polyadic decomposition of nonnegative third order tensors. Signal Process 91(9):2159–2171

    MATH  Google Scholar 

  • Sang C (2019) A new Brauer-type Z-eigenvalue inclusion set fortensors. Numer Algorithms 32:781–794

    MATH  Google Scholar 

  • Sang C, Chen Z (2019) Z-Eigenvalue localization sets for even order tensors and their applications. Acta Appl Math. https://doi.org/10.1007/s10440-019-00300-1

    Google Scholar 

  • Song Y, Qi L (2013) Spectral properties of positively homogeneous operators induced by higher order tensors. SIAM J Matrix Anal Appl 34(4):1581–1595

    MathSciNet  MATH  Google Scholar 

  • Sørensen M, De Lathauwer L (2015) New uniqueness conditions for the canonical polyadic decomposition of third-order tensors. SIAM J Matrix Anal Appl 36(4):1381–1403

    MathSciNet  MATH  Google Scholar 

  • Wang X, Che M, Wei Y (2019) Neural networks based approach solving multi-linear systems with M-tensors. Neurocomputing 351:33–42

    Google Scholar 

  • Wang W, Chen H, Wang Y (2020) A new C-eigenvalue interval for piezoelectric-type tensors. Appl Math Lett. https://doi.org/10.1016/j.aml.2019.106035

    MathSciNet  MATH  Google Scholar 

  • Xiong L, Liu J (2020) Z-eigenvalue inclusion theorem of tensors and the geometric measure of entanglement of multipartite pure states. Comput Appl Math 39:135. https://doi.org/10.1007/s40314-020-01166-y

    MathSciNet  MATH  Google Scholar 

  • Zhang T, Golub GH (2001) Rank-one approximation to high order tensors. SIAM J Matrix Anal Appl 23(2):534–550

    MathSciNet  MATH  Google Scholar 

  • Zhang L, Qi L, Zhou G (2014) M-tensors and some applications. SIAM J Matrix Anal Appl 35(2):437–452

    MathSciNet  MATH  Google Scholar 

  • Zhao J (2017) Sang C (2017) An eigenvalue localization set for tensors and its applications. J Inequal Appl 1:1–9

    Google Scholar 

Download references

Acknowledgements

The authors are grateful to the anonymous referees for their helpful suggestions and comments. This work is supported by Science and Technology Foundation of Guizhou Province ([2020]1Z002), Guizhou Education Department Youth Science and Technology Talents Growth Project (Grant no. QJHKYZ [2016]066) and National Natural Science Foundation of China (Grant no. 11501141).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zhen Chen.

Additional information

Communicated by Yimin Wei.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, S., Chen, Z., Li, C. et al. Eigenvalue bounds of third-order tensors via the minimax eigenvalue of symmetric matrices. Comp. Appl. Math. 39, 217 (2020). https://doi.org/10.1007/s40314-020-01245-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-020-01245-0

Keywords

Mathematics Subject Classification

Navigation