Properties of Some Classes of Structured Tensors

  • Yisheng SongEmail author
  • Liqun Qi


In this paper, we extend some classes of structured matrices to higher-order tensors. We discuss their relationships with positive semi-definite tensors and some other structured tensors. We show that every principal sub-tensor of such a structured tensor is still a structured tensor in the same class, with a lower dimension. The potential links of such structured tensors with optimization, nonlinear equations, nonlinear complementarity problems, variational inequalities and the non-negative tensor theory are also discussed.


P tensor \(\hbox {P}_0\) tensor B tensor \(\hbox {B}_0\) tensor Principal sub-tensor Eigenvalues 

Mathematics Subject Classification (2010)

47H15 47H12 34B10 47A52 47J10 47H09 15A48 47H07 



The authors would like to thank the anonymous referees for their valuable suggestions which helped us to improve this manuscript.


  1. 1.
    Fiedler, M., Pták, V.: On matrices with non-positive off-diagonal elements and positive principal minors. Czechoslov. Math. J. 12, 163–172 (1962)Google Scholar
  2. 2.
    Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Boston (1992)zbMATHGoogle Scholar
  3. 3.
    Facchinei, F., Pang, J.S.: Finite Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)Google Scholar
  4. 4.
    Qi, L., Sun, D., Zhou, G.: A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Math. Program. 87, 1–35 (2000)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Bose, N.K., Modaress, A.R.: General procedure for multivariable polynomial positivity with control applications. IEEE Trans. Autom. Control 21, 596–601 (1976)Google Scholar
  6. 6.
    Hasan, M.A., Hasan, A.A.: A procedure for the positive definiteness of forms of even-order. IEEE Trans. Autom. Control 41, 615–617 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Jury, E.I., Mansour, M.: Positivity and nonnegativity conditions of a quartic equation and related problems. IEEE Trans. Automat. Control 26, 444–451 (1981)Google Scholar
  8. 8.
    Wang, F., Qi, L.: Comments on ‘Explicit criterion for the positive definiteness of a general quartic form’. IEEE Trans. Autom. Control 50, 416–418 (2005)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)CrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, Y., Dai, Y., Han, D., Sun, W.: Positive semidefinite generalized diffusion tensor imaging via quadratic semidefinite programming. SIAM J. Imaging Sci. 6, 1531–1552 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Hu, S., Huang, Z., Ni, H., Qi, L.: Positive definiteness of diffusion kurtosis imaging. Inverse Probl. Imaging 6, 57–75 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Qi, L., Yu, G., Wu, E.X.: Higher order positive semi-definite diffusion tensor imaging. SIAM J. Imaging Sci. 3, 416–433 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Qi, L., Yu, G., Xu, Y.: Nonnegative diffusion orientation distribution function. J. Math. Imaging Vis. 45, 103–113 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Hu, S., Qi, L.: Algebraic connectivity of an even uniform hypergraph. J. Comb. Optim. 24, 564–579 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Li, G., Qi, L., Yu, G.: The Z-eigenvalues of a symmetric tensor and its application to spectral hypergraph theory. Numer. Linear Algebra Appl. 20, 1001–1029 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Qi, L.: H\(^+\)-eigenvalues of Laplacian and signless Laplacian tensors. Commun. Math. Sci. 12, 1045–1064 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Peña, J.M.: A class of P-matrices with applications to the localization of the eigenvalues of a real matrix. SIAM J. Matrix Anal. Appl. 22, 1027–1037 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Peña, J.M.: On an alternative to Gerschgorin circles and ovals of Cassini. Numerische Mathematik 95, 337–345 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Zhang, L., Qi, L., Zhou, G.: M-tensors and some applications. SIAM J. Matrix Anal. Appl. 35(2), 437–452 (2014)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Hu, S., Qi, L.: 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)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Hu, S., Qi, L., Shao, J.: Cored hypergraphs, power hypergraphs and their Laplacian eigenvalues. Linear Algebra Appl. 439, 2980–2998 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Hu, S., Qi, L., Xie, J.: The largest Laplacian and signless Laplacian eigenvalues of a uniform hypergraph. arXiv:1304.1315 (2013)
  23. 23.
    Qi, L., Shao, J., Wang, Q.: Regular uniform hypergraphs, \(s\)-cycles, \(s\)-paths and their largest Laplacian H-eigenvalues. Linear Algebra Appl. 443, 215–227 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Mathias, R., Pang, J.S.: Error bounds for the linear complementarity problem with a P-matrix. Linear Algebra Appl. 132, 123–136 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Mathias, R.: An improved bound for a fundamental constant associated with a P-matrix. Appl. Math. Lett. 2, 297–300 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Xiu, N., Zhang, J.: A characteristic quantity of P-matrices. Appl. Math. Lett. 15, 41–46 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    García-Esnaola, M., Peña, J.M.: Error bounds for linear complementarity problems for B-matrices. Appl. Math. Lett. 22, 1071–1075 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Song, Y., Qi, L.: Spectral properties of positively homogeneous operators induced by higher order tensors. SIAM J. Matrix Anal. Appl. 34, 1581–1595 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Ding, W., Qi, L., Wei, Y.: M-tensors and nonsingular M-tensors. Linear Algebra Appl. 439, 3264–3278 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Ortega, J.M., Rheinboldt, W.C.: Iterative Solutions of Nonlinear Equations in Several Variables. Academic Press, New York (1970)Google Scholar
  31. 31.
    Cichocki, A., Zdunek, R., Phan, A.H., Amari, S.: Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multiway Data Analysis and Blind Source Separation. Wiley, New York (2009)CrossRefGoogle Scholar
  32. 32.
    Chang, K.C., Qi, L., Zhang, T.: A survey on the spectral theory of nonnegative tensors. Numer. Linear Algebra Appl. 20, 891–912 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Qi, L., Xu, C., Xu, Y.: Nonnegative tensor factorization, completely positive tensors and an hierarchically elimination algorithm. to appear. SIAM J. Matrix Anal. Appl.Google Scholar
  34. 34.
    Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadephia (1994)CrossRefzbMATHGoogle Scholar
  35. 35.
    Qi, L., Song, Y.: An even order symmetric B tensor is positive definite. Linear Algebra Appl. 457, 303–312 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Yuan, P., You, L.: Some remarks on P, P\(_0\), B and B\(_0\) tensors. arXiv:1402.1288 (2014)
  37. 37.
    Song, Y., Qi, L.: Infinite and finite dimensional Hilbert tensors. Linear Algebra Appl. 451, 1–14 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Chen, H., Qi, L.: Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors. arXiv:1405.6363 (2014)

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceHenan Normal UniversityXinxiangPeople’s Republic of China
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHung Hom, KowloonHong Kong

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