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

  • Caili Sang
  • Zhen ChenEmail author
Article
  • 14 Downloads

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.

Keywords

Nonnegative tensors \(Z\)-eigenvalues \(Z\)-spectral radius Localization sets Positive definiteness 

Mathematics Subject Classification (2010)

15A18 15A42 15A69 

Notes

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.

References

  1. 1.
    Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 4, 1302–1324 (2005) MathSciNetCrossRefGoogle Scholar
  2. 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. 3.
    Qi, L., Chen, H., Chen, Y.: Tensor Eigenvalues and Their Applications. Springer, Singapore (2018) CrossRefGoogle Scholar
  4. 4.
    Hsu, J.C., Meyer, A.U.: Modern Control Principles and Applications. McGraw-Hill, New York (1968) zbMATHGoogle Scholar
  5. 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) CrossRefGoogle Scholar
  6. 6.
    Bose, N.K., Newcomb, R.W.: Tellegons theorem and multivariate realizability theory. Int. J. Electron. 36, 417–425 (1974) CrossRefGoogle Scholar
  7. 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) MathSciNetCrossRefGoogle Scholar
  8. 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) MathSciNetCrossRefGoogle Scholar
  9. 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) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Li, C., Li, Y., Kong, X.: New eigenvalue inclusion sets for tensors. Numer. Linear Algebra Appl. 21, 39–50 (2014) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Li, C., Chen, Z., Li, Y.: A new eigenvalue inclusion set for tensors and its applications. Linear Algebra Appl. 481, 36–53 (2015) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Li, C., Zhou, J., Li, Y.: A new Brauer-type eigenvalue localization set for tensors. Linear Multilinear Algebra 64(4), 727–736 (2016) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Li, C., Jiao, A., Li, Y.: An \(S\)-type eigenvalue localization set for tensors. Linear Algebra Appl. 493, 469–483 (2016) MathSciNetCrossRefGoogle Scholar
  14. 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 CrossRefGoogle Scholar
  15. 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 CrossRefGoogle Scholar
  16. 16.
    Wang, G., Zhou, G., Caccetta, L.: \(Z\)-eigenvalue inclusion theorems for tensors. Discrete Contin. Dyn. Syst., Ser. B 22, 187–198 (2017) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Zhao, J.: A new \(Z\)-eigenvalue localization set for tensors. J. Inequal. Appl. 2017, 85 (2017) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhao, J., Sang, C.: Two new eigenvalue localization sets for tensors and theirs applications. Open Math. 16, 1267–1276 (2017) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Wang, Y.N., Wang, G.: Two \(S\)-type \(Z\)-eigenvalue inclusion sets for tensors. J. Inequal. Appl. 2017, 152 (2017) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sang, C.: A new Brauer-type Z-eigenvalue inclusion set for tensors. Numer. Algorithms 80, 781–794 (2019) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Merris, R.: Combinatorics, 2nd edn. Wiley, New York (2003) CrossRefGoogle Scholar
  22. 22.
    Marsli, R., Hall, F.J.: On bounding the eigenvalues of matrices with constant row-sums. Linear Multilinear Algebra 67(4), 672–684 (2019) MathSciNetCrossRefGoogle Scholar
  23. 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) MathSciNetCrossRefGoogle Scholar
  24. 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) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zhang, T., Golub, G.H.: Rank-one approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 23(2), 534–550 (2001) MathSciNetCrossRefGoogle Scholar
  26. 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. 27.
    Qi, L., Yu, G., Wu, E.X.: Higher order positive semidefinite diffusion tensor imaging. SIAM J. Imaging Sci. 3(3), 416–433 (2010) MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kolda, T.G., Mayo, J.R.: Shifted power method for computing tensor eigenpairs. SIAM J. Matrix Anal. Appl. 32(4), 1095–1124 (2011) MathSciNetCrossRefGoogle Scholar
  29. 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) MathSciNetCrossRefGoogle Scholar
  30. 30.
    Devore, R.A., Temlyakov, V.N.: Some remarks on greedy algorithms. Adv. Comput. Math. 5, 173–187 (1996) MathSciNetCrossRefGoogle Scholar
  31. 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) MathSciNetCrossRefGoogle Scholar
  32. 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) MathSciNetCrossRefGoogle Scholar
  33. 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) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Qi, L.: The best rank-one approximation ratio of a tensor space. SIAM J. Matrix Anal. Appl. 32(2), 430–442 (2011) MathSciNetCrossRefGoogle Scholar
  35. 35.
    Liu, Q., Li, Y.: Bounds for the \(Z\)-eigenpair of general nonnegative tensors. Open Math. 14, 181–194 (2016) MathSciNetCrossRefGoogle Scholar
  36. 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) MathSciNetCrossRefGoogle Scholar
  37. 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) MathSciNetCrossRefGoogle Scholar
  38. 38.
    Li, W., Liu, D., Vong, S.-W.: \(Z\)-eigenpair bounds for an irreducible nonnegative tensor. Linear Algebra Appl. 483, 182–199 (2015) MathSciNetCrossRefGoogle Scholar
  39. 39.
    He, J.: Bounds for the largest eigenvalue of nonnegative tensors. J. Comput. Anal. Appl. 20, 1290–1301 (2016) MathSciNetzbMATHGoogle Scholar
  40. 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) CrossRefGoogle Scholar
  41. 41.
    He, J., Huang, T.-Z.: Upper bound for the largest \(Z\)-eigenvalue of positive tensors. Appl. Math. Lett. 38, 110–114 (2014) MathSciNetCrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesGuizhou Normal UniversityGuiyangP.R. China
  2. 2.College of Data Science and Information EngineeringGuizhou Minzu UniversityGuiyangP.R. China

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