Skip to main content
SpringerLink
Log in

\(Z\)-Eigenvalue Localization Sets for Even Order Tensors and Their Applications

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

Firstly, a new Geršgorin-type \(Z\)-eigenvalue localization set with parameters for even order tensors is presented. As an application, some sufficient conditions for the positive (semi-)definiteness of even order real symmetric tensors are obtained. Secondly, by selecting appropriate parameters an optimal set is obtained and proved to be tighter than some existing results. Thirdly, as another application, new upper bounds for the \(Z\)-spectral radius of even order weakly symmetric nonnegative tensors are obtained. Finally, numerical examples are given to verify the theoretical results.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 4, 1302–1324 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Lim, L.H.: Singular values and eigenvalues of tensors: a variational approach. In: CAMSAP’05: Proceeding of the IEEE International Workshop on Computational Advances in MultiSensor Adaptive Processing, pp. 129–132 (2005)

    Google Scholar 

  3. Qi, L., Chen, H., Chen, Y.: Tensor Eigenvalues and Their Applications. Springer, Singapore (2018)

    Book  MATH  Google Scholar 

  4. Hsu, J.C., Meyer, A.U.: Modern Control Principles and Applications. McGraw-Hill, New York (1968)

    MATH  Google Scholar 

  5. Bose, N.K., Kamat, P.S.: Algorithm for stability test of multidimensional filters. IEEE Trans. Acoust. Speech Signal Process. ASSP. 22, 307–314 (1974)

    Article  Google Scholar 

  6. Bose, N.K., Newcomb, R.W.: Tellegons theorem and multivariate realizability theory. Int. J. Electron. 36, 417–425 (1974)

    Article  Google Scholar 

  7. Anderson, B.D.O., Bose, N.K., Jury, E.I.: Output feedback stabilization and related problems-solutions via decision methods. IEEE Trans. Autom. Control AC20, 53–66 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cui, L.B., Li, M.H., Song, Y.: Preconditioned tensor splitting iterations method for solving multi-linear systems. Appl. Math. Lett. 96, 89–94 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, C., Li, Y.: An eigenvalue localization set for tensors with applications to determine the positive (semi-)definitenss of tensors. Linear Multilinear Algebra 64(4), 587–601 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Li, C., Chen, Z., Li, Y.: A new eigenvalue inclusion set for tensors and its applications. Linear Algebra Appl. 481, 36–53 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, C., Zhou, J., Li, Y.: A new Brauer-type eigenvalue localization set for tensors. Linear Multilinear Algebra 64(4), 727–736 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, C., Jiao, A., Li, Y.: An \(S\)-type eigenvalue localization set for tensors. Linear Algebra Appl. 493, 469–483 (2016)

    MathSciNet  MATH  Google Scholar 

  14. He, J., Liu, Y., Xu, G.: \(Z\)-eigenvalues-based sufficient conditions for the positive definiteness of fourth-order tensors. Bull. Malays. Math. Sci. Soc. (2019). https://doi.org/10.1007/s40840-019-00727-7

    MATH  Google Scholar 

  15. Zhao, J.: \(E\)-eigenvalue localization sets for fourth-order tensors. Bull. Malays. Math. Sci. Soc. (2019). https://doi.org/10.1007/s40840-019-00768-y

    Google Scholar 

  16. Wang, G., Zhou, G., Caccetta, L.: \(Z\)-eigenvalue inclusion theorems for tensors. Discrete Contin. Dyn. Syst., Ser. B 22, 187–198 (2017)

    MathSciNet  MATH  Google Scholar 

  17. Zhao, J.: A new \(Z\)-eigenvalue localization set for tensors. J. Inequal. Appl. 2017, 85 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Zhao, J., Sang, C.: Two new eigenvalue localization sets for tensors and theirs applications. Open Math. 16, 1267–1276 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, Y.N., Wang, G.: Two \(S\)-type \(Z\)-eigenvalue inclusion sets for tensors. J. Inequal. Appl. 2017, 152 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Sang, C.: A new Brauer-type Z-eigenvalue inclusion set for tensors. Numer. Algorithms 80, 781–794 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Merris, R.: Combinatorics, 2nd edn. Wiley, New York (2003)

    Book  MATH  Google Scholar 

  22. Marsli, R., Hall, F.J.: On bounding the eigenvalues of matrices with constant row-sums. Linear Multilinear Algebra 67(4), 672–684 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kofidis, E., Regalia, P.A.: On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J. Matrix Anal. Appl. 23, 863–884 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lathauwer, L.D., Moor, B.D., Vandewalle, J.: On the best rank-1 and rank-(\(R _{1},R_{2},\ldots ,R_{N}\)) approximation of higer-order tensors. SIAM J. Matrix Anal. Appl. 21(4), 1324–1342 (2000)

    MathSciNet  MATH  Google Scholar 

  25. Zhang, T., Golub, G.H.: Rank-one approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 23(2), 534–550 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Bloy, L., Verma, R.: On computing the underlying fiber directions from the diffusion orientation distribution function. In: Medical Image Computing and Computer-Assisted Intervention, vol. 5241, pp. 1–8. Springer, Berlin (2008)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Kolda, T.G., Mayo, J.R.: Shifted power method for computing tensor eigenpairs. SIAM J. Matrix Anal. Appl. 32(4), 1095–1124 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Qi, L.: Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines. J. Symb. Comput. 41, 1309–1327 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Devore, R.A., Temlyakov, V.N.: Some remarks on greedy algorithms. Adv. Comput. Math. 5, 173–187 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  31. Falco, A., Nouy, A.: A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach. J. Math. Anal. Appl. 376, 469–480 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, Y., Qi, L.: On the successive supersymmetric rank-1 decomposition of higher-order supersymmetric tensors. Numer. Linear Algebra Appl. 14, 503–519 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ammar, A., Chinesta, F., Falcó, A.: On the convergence of a greedy rank-one update algorithm for a class of linear systems. Arch. Comput. Methods Eng. 17, 473–486 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Qi, L.: The best rank-one approximation ratio of a tensor space. SIAM J. Matrix Anal. Appl. 32(2), 430–442 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Liu, Q., Li, Y.: Bounds for the \(Z\)-eigenpair of general nonnegative tensors. Open Math. 14, 181–194 (2016)

    MathSciNet  MATH  Google Scholar 

  36. Chang, K.C., Pearson, K.J., Zhang, T.: Some variational principles for \(Z\)-eigenvalues of nonnegative tensors. Linear Algebra Appl. 438, 4166–4182 (2013)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  39. He, J.: Bounds for the largest eigenvalue of nonnegative tensors. J. Comput. Anal. Appl. 20, 1290–1301 (2016)

    MathSciNet  MATH  Google Scholar 

  40. He, J., Liu, Y.-M., Ke, H., Tian, J.-K., Li, X.: Bounds for the \(Z\)-spectral radius of nonnegative tensors. SpringerPlus 5, 1727 (2016)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referees and Editors-in-Chief John King, Benoît Perthame for their comments and suggestions. This work is supported by Science and Technology Projects of Education Department of Guizhou Province (Grant No. KY[2015]352); Science and Technology Top-notch Talents Support Project of Education Department of Guizhou Province (Grant No. QJHKYZ [2016]066); National Natural Science Foundations of China (Grant No. 11501141) and Natural Science Foundation of Guizhou Minzu University.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou, 550025, P.R. China

    Caili Sang & Zhen Chen

  2. College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou, 550025, P.R. China

    Caili Sang

Authors
  1. Caili Sang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhen Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhen Chen.

Additional information

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

Sang, C., Chen, Z. \(Z\)-Eigenvalue Localization Sets for Even Order Tensors and Their Applications. Acta Appl Math 169, 323–339 (2020). https://doi.org/10.1007/s10440-019-00300-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-019-00300-1

Keywords

Mathematics Subject Classification (2010)

Search

Navigation