Some criteria for identifying strong \(\mathcal {H}\)-tensors

Original Paper
  • 29 Downloads

Abstract

In this paper, we present some new criteria only depending on the elements of the given tensors for judging strong \(\mathcal {H}\)-tensors which cannot be identified by some existing criteria in Li et al. (J. Comput. Appl. Math. 255, 1–4, 2014) and Zhao et al. (Front. Math. China 11, 661–678, 2016). Some new necessary and sufficient conditions of strong \(\mathcal {H}\)-tensors are also provided. As an application, some sufficient conditions for identifying the positive definiteness of a class of multivariate forms are obtained. These facts are well illustrated by some numerical examples.

Keywords

Strong \(\mathcal {H}\)-tensors Criteria Multivariate form Positive definiteness 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The authors would like to thank two anonymous referees for their valuable suggestions and constructive comments that improved the quality of this paper.

References

  1. 1.
    Chang, K.C., Pearson, K., Zhang, T.: Perron-Frobenius theorem for nonnegative tensors. Commun. Math. Sci. 6, 507–520 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Qi, L.Q.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Lathauwer, L.D., Moor, B.D., Vandewalle, J.: On the best rank-1 and rank-(r 1, r 2,⋅⋅⋅, r n) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 21, 1324–1342 (2000)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Hillar, C.J., Lim, L.H.: Most tensor problems are NP-hard. J. ACM 60, 1–39 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ding, W.Y., Qi, L.Q., Wei, Y.M.: \(\mathcal {M}\)-tensors and nonsingular \(\mathcal {M}\)-tensors. Linear Algebra Appl. 439, 3264–3278 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, H.B., Qi, L.Q.: Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors. J. Ind. Manag. Optim. 11, 1263–1274 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    He, J., Huang, T.Z.: Inequalities for \(\mathcal {M}\)-tensors. J. Inequal. Appl. 114, 1–9 (2014)MathSciNetGoogle Scholar
  8. 8.
    Qi, L.Q.: Hankel tensors: associated Hankel matrices and Vandermonde decomposition. Commun. Math. Sci. 13, 113–125 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Song, Y.S., Qi, L.Q.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165, 854–873 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Yuan, P.Z., You, L.H.: Some remarks on \(\mathcal {P}\), \(\mathcal {P}_0\), \(\mathcal {B}\) and \(\mathcal {B}_0\) tensors. Linear Algebra Appl. 459, 511–521 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Zhang, L.P., Qi, L.Q., Zhou, G.L.: \(\mathcal {M}\)-tensors and some applications. SIAM J. Matrix Anal. Appl. 35, 437–452 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Qi, L.Q., Song, Y.S.: An even order symmetric \(\mathcal {B}\)-tensor is positive definite. Linear Algebra Appl. 457, 303–312 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chen, H.B., Li, G.Y., Qi, L.Q.: Further results on Cauchy tensors and Hankel tensors. Appl. Math. Comput. 275, 50–62 (2016)MathSciNetGoogle Scholar
  14. 14.
    Gan, T.B., Huang, T.Z.: Simple criteria for nonsingular H-matrices. Linear Algebra Appl. 374, 317–326 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Liu, J.Z., He, A.Q.: Simple criteria for generalized diagonally dominant matrices. Int. J. Comput. Math. 85, 1065–1072 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Cvetković, L., Kostić, V.: New criteria for identifying H-matrices. J. Comput. Appl. Math. 180, 265–278 (2005)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Huang, T.Z., Evans, D.J.: Generalized doubly diagonally dominant matrices. Int. J. Comput. Math. 81, 863–870 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Gao, Y.M., Wang, X.H.: Criteria for generalized diagonally dominant matrices and M-matrices. Linear Algebra Appl. 169, 257–268 (1992)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Li, C.Q., Wang, F., Zhao, J.X., Li, Y.T.: Criterions for the positive definiteness of real supersymmetric tensors. J. Comput. Appl. Math. 255, 1–14 (2014)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Wang, F., Sun, D.S., Zhao, J.X., Li, C.Q.: New practical criteria for \(\mathcal {H}\)-tensors and its application. Linear Multilinear Algebra 65, 269–283 (2017)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Zhao, R.J., Gao, L., Liu, Q.L., Li, Y.T.: Criterions for identifying \(\mathcal {H}\)-tensors. Front. Math. China 11, 661–678 (2016)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Li, Y.T., Liu, Q.L., Qi, L.Q.: Programmable criteria for strong \(\mathcal {H}\)-tensors. Numer. Algorithm 74, 199–221 (2017)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kannan, M.R., Shaked-Monderer, N., Berman, A.: Some properties of strong \(\mathcal {H}\)-tensors and general \(\mathcal {H}\)-tensors. Linear Algebra Appl. 476, 42–55 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Wang, X.Z., Wei, Y.M.: \(\mathcal {H}\)-tensors and nonsingular \(\mathcal {H}\)-tensors. Front. Math. China 11, 557–575 (2016)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lim, L.H.: Singular values and eigenvalues of tensors: a variational approach. In: CAMSAP’05: Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, vol. 1, pp. 129–132 (2005)Google Scholar
  26. 26.
    Ni, Q., Qi, L.Q., Wang, F.: An eigenvalue method for testing positive definiteness of a multivariate form. IEEE Trans. Automat. Control 53, 1096–1107 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Bose, N.K., Modarressi, A.R.: General procedure for multivariable polynomial positivity with control applications. IEEE Trans. Automat. Control 21, 696–701 (1976)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Bose, N.K., Kamat, P.S.: Algorithm for stability test of multidimensional filters. IEEE Trans. Acoust. Speech Signal Process. 22, 307–314 (1974)CrossRefGoogle Scholar
  29. 29.
    Chen, Y.N., Dai, Y.H., Han, D., Sun, W.Y.: Positive semidefinite generalized diffusion tensor imaging via quadratic semidefinite programming. SIAM J. Imaging Sci. 6, 1531–1552 (2013)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Hu, S.L., Huang, Z.H., Ni, H.Y., Qi, L.Q.: Positive definiteness of diffusion kurtosis imaging. Inverse Probl. Imag. 6, 57–75 (2012)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Shor, N.Z.: Nondifferentiable Optimization and Polynomial Problems. Kluwer Academic Publishers, Boston (1998)CrossRefMATHGoogle Scholar
  32. 32.
    Hu, S.L., Qi, L.Q.: Algebraic connectivity of an even uniform hypergraph. J. Comb. Optim. 24, 564–579 (2012)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Hu, S.L., Qi, L.Q.: The eigenvectors associated with the zero eigenvalues of the Laplacian and signless Laplacian tensors of a uniform hypergraph. Discrete Appl. Math. 169, 140–151 (2014)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Li, G.Y., Qi, L.Q., Yu, G.H.: The Z-eigenvalues of a symmetric tensor and its application to spectral hypergraph theory. Numer. Linear Algebra Appl. 20, 1001–1029 (2013)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Li, C.Q., Li, Y.T., Kong, X.: New eigenvalue inclusion sets for tensors. Numer. Linear Algebra Appl. 21, 39–50 (2014)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Li, C.Q., Li, Y.T.: An eigenvalue localization set for tensors with applications to determine the positive (semi-)definiteness of tensors. Linar Multilinear Algebra 64, 587–601 (2016)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Qi, L.Q., Luo, Z.Y.: Tensor Analysis: Spectral Theory and Special Tensors. SIAM, Philadelphia (2017)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China

Personalised recommendations