Advertisement

Z-eigenvalues based structured tensors: \(\mathcal {M}_z\)-tensors and strong \(\mathcal {M}_z\)-tensors

  • Changxin Mo
  • Chaoqian Li
  • Xuezhong Wang
  • Yimin WeiEmail author
Article

Abstract

The positive (semi-)definiteness of even-order tensors has been widely studied in these years due to its applications in various aspects, such as spectral hypergraph theory, automatic control, polynomial theory, stochastic process, magnetic resonance imaging and so on. It has been shown that \(\mathcal {M}\)-tensors, \(\mathcal {B}\)-tensors, \(\mathcal {H}\)-tensors, Hilbert tensors and stochastic tensors can be positive definite under proper conditions. However, there are still many positive definite tensors that can not be determined by the above criteria. In this paper, we provide a new class of positive definite tensors whose non-diagonal entries can be positive compared to (strong) \(\mathcal {M}\)-tensors, and we call it strong \(\mathcal {M}_z\)-tensors, which can arise from discretizing differential equations, since it is based on Z-eigenvalues rather than H-eigenvalues traditionally. Moreover, we show that an even-order (strong) \(\mathcal {M}\)-tensor must be an (a strong) \(\mathcal {M}_z\)-tensor, which reflects the inclusion relationship between even-order \(\mathcal {M}\)-tensors and \(\mathcal {M}_z\)-tensors. We also introduce (strong) \(\mathcal {H}_z\)-tensors, as a generalization of (strong) \(\mathcal {M}_z\)-tensors, and its positive semi-definiteness (positive definiteness) has been studied. Finally, some conditions are given for a tensor to be an (a strong) \(\mathcal {M}_z\)-tensor and we use it to study the stability of a high-order nonlinear system.

Keywords

\(\mathcal {M}_z\)-tensor Z-eigenvalue \(\mathcal {H}_z\)-tensor \(\mathcal {M}\)-tensor Positive definiteness Stability of nonlinear system 

Mathematics Subject Classification

15A18 15A69 

Notes

Acknowledgements

The authors are indebted to the editor and two referees for their constructive comments and valuable suggestions, which greatly improved the original manuscript of this paper. Also, the first author would like to thank Dr. Maolin Che, Dr. Weiyang Ding and Dr. Liping Zhang for their useful discussions and valuable suggestions.

References

  1. Berman A, Plemmons RJ (1994) Nonnegative matrices in the mathematical sciences. SIAM, Philadelphia (classics ed)CrossRefzbMATHGoogle Scholar
  2. Bose N, Kamat P (1974) Algorithm for stability test of multidimensional filters. IEEE Trans Acoust Speech Signal Process 22:307–314CrossRefGoogle Scholar
  3. Bose N, Modarressi A (1976) General procedure for multivariable polynomial positivity test with control applications. IEEE Trans Autom Control 21:696–701CrossRefMathSciNetzbMATHGoogle Scholar
  4. Bulò SR, Pelillo M (2009) New bounds on the clique number of graphs based on spectral hypergraph theory. In: Learning and intelligent optimization, third international conference, LION 3, Trento, Italy, January 14–18. Selected Papers 2009:45–58Google Scholar
  5. Chang KC, Pearson KJ, Zhang T (2008) Perron-Frobenius theorem for nonnegative tensors. Commun Math Sci 6:507–520CrossRefMathSciNetzbMATHGoogle Scholar
  6. Chang KC, Pearson KJ, Zhang T (2009) On eigenvalue problems of real symmetric tensors. J Math Anal Appl 350:416–422CrossRefMathSciNetzbMATHGoogle Scholar
  7. Chang KC, Pearson KJ, Zhang T (2013) Some variational principles for \(Z\)-eigenvalues of nonnegative tensors. Linear Algebra Appl 438:4166–4182CrossRefMathSciNetzbMATHGoogle Scholar
  8. Che M, Cichocki A, Wei Y (2017) Neural networks for computing best rank-one approximations of tensors and its applications. Neurocomputing 267:114–133CrossRefGoogle Scholar
  9. Che M, Qi L, Wei Y (2016) Positive-definite tensors to nonlinear complementarity problems. J Optim Theory Appl 168:475–487CrossRefMathSciNetzbMATHGoogle Scholar
  10. Che M, Qi L, Wei Y (2019) Stochastic \({R}_0\) tensors to stochastic tensor complementarity problems. Optim Lett 13:261–279CrossRefMathSciNetzbMATHGoogle Scholar
  11. Che M, Wei Y (2019) Randomized algorithms for the approximations of Tucker and the tensor train decompositions. Adv Comput Math 45:395–428CrossRefMathSciNetzbMATHGoogle Scholar
  12. Chen L, Han L, Zhou L (2016) Computing tensor eigenvalues via homotopy methods. SIAM J Matrix Anal Appl 37:290–319CrossRefMathSciNetzbMATHGoogle Scholar
  13. Chen Y, Dai Y, Han D, Sun W (2013) Positive semidefinite generalized diffusion tensor imaging via quadratic semidefinite programming. SIAM J Imaging Sci 6:1531–1552CrossRefMathSciNetzbMATHGoogle Scholar
  14. Chen Y, Qi L, Wang Q (2016) Positive semi-definiteness and sum-of-squares property of fourth order four dimensional Hankel tensors. J Comput Appl Math 302:356–368CrossRefMathSciNetzbMATHGoogle Scholar
  15. Cichocki A, Zdunek R, Phan AH, Amari S-i (2009) Nonnegative matrix and tensor factorizations: applications to exploratory multi-way data analysis and blind source separation. Wiley, New YorkCrossRefGoogle Scholar
  16. Comon P, Lim LH (2011) Sparse representations and low-rank tensor approximation. I3S/RR-2011-02-FR, En cours de soumission sous le titre ”Multilinear identification”Google Scholar
  17. Deng C, Li H, Bu C (2018) Brauer-type eigenvalue inclusion sets of stochastic/irreducible tensors and positive definiteness of tensors. Linear Algebra Appl 556:55–69CrossRefMathSciNetzbMATHGoogle Scholar
  18. Ding J, Zhou A (2009) Nonnegative matrices, positive operators, and applications. World Scientific, River Edge, NJCrossRefzbMATHGoogle Scholar
  19. Ding W, Hou Z, Wei Y (2016) Tensor logarithmic norm and its applications. Numerical Linear Algebra Appl 23:989–1006CrossRefMathSciNetzbMATHGoogle Scholar
  20. Ding W, Luo Z, Qi L (2018) \(P\)-tensors, \(P_0\)-tensors, and their applications. Linear Algebra and its Applications 555:336–354CrossRefMathSciNetzbMATHGoogle Scholar
  21. Ding W, Qi L, Wei Y (2013) \(\cal{M}\)-tensors and nonsingular \(\cal{M}\)-tensors. Linear Algebra Appl 439:3264–3278CrossRefMathSciNetzbMATHGoogle Scholar
  22. Ding W, Wei Y (2015) Generalized tensor eigenvalue problems. SIAM J Matrix Anal Appl 36:1073–1099CrossRefMathSciNetzbMATHGoogle Scholar
  23. Ding W, Wei Y (2016) Solving multi-linear systems with \(mathcal M \)-tensors. J Sci Comput 68:689–715CrossRefMathSciNetzbMATHGoogle Scholar
  24. Hiriart-Urruty J-B, Seeger A (2010) A variational approach to copositive matrices. SIAM Rev 52:593–629CrossRefMathSciNetzbMATHGoogle Scholar
  25. Hu S, Huang Z-H, Ni H-Y, Qi L (2012) Positive definiteness of diffusion kurtosis imaging. Inverse Problems Imaging 6:57–75CrossRefMathSciNetzbMATHGoogle Scholar
  26. Huang Z-H, Qi L (2018) Positive definiteness of paired symmetric tensors and elasticity tensors. J Comput Appl Math 338:22–43CrossRefMathSciNetzbMATHGoogle Scholar
  27. Kannan MR, Shaked-Monderer N, Berman A (2015) Some properties of strong \(\cal{H}\)-tensors and general \(\cal{H}\)-tensors. Linear Algebra Appl 476:42–55CrossRefMathSciNetzbMATHGoogle Scholar
  28. Kofidis E, Regalia P (2002) On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J Matrix Anal Appl 23:863–884CrossRefMathSciNetzbMATHGoogle Scholar
  29. Kolda TG, Bader BW (2009) Tensor decompositions and applications. SIAM Rev 51:455–500CrossRefMathSciNetzbMATHGoogle Scholar
  30. Kolda TG, Mayo JR (2010) Shifted power method for computing tensor eigenpairs. SIAM J Matrix Anal Appl 32:1095–1124CrossRefMathSciNetzbMATHGoogle Scholar
  31. Lathauwer LD, Moor BD, Vandewalle J (2006) A multilinear singular value decomposition. SIAM J Matrix Anal Appl 21:1253–1278CrossRefMathSciNetzbMATHGoogle Scholar
  32. Li C, Li Y (2016) An eigenvalue localization set for tensors with applications to determine the positive (semi-) definiteness of tensors. Linear Multilinear Algebra 64:587–601CrossRefMathSciNetzbMATHGoogle Scholar
  33. Li C, Wang F, Zhao J, Zhu Y, Li Y (2014) Criterions for the positive definiteness of real supersymmetric tensors. J Comput Appl Math 255:1–14CrossRefMathSciNetzbMATHGoogle Scholar
  34. Li D-H, Xie S, Xu H-R (2017) Splitting methods for tensor equations. Numer Linear Algebra Appl 24:e2102CrossRefMathSciNetzbMATHGoogle Scholar
  35. Li X, Ng MK (2015) Solving sparse non-negative tensor equations: algorithms and applications. Front Math China 10:649–680CrossRefMathSciNetzbMATHGoogle Scholar
  36. Lim LH (2006) Singular values and eigenvalues of tensors: a variational approach. In: IEEE international workshop on computational advances in multi-sensor adaptive processing, pp. 129–132Google Scholar
  37. Liu Y, Zhou G, Ibrahim NF (2010) An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor. J Comput Appl Math 235:286–292CrossRefMathSciNetzbMATHGoogle Scholar
  38. Moakher M (2006) On the averaging of symmetric positive-definite tensors. J Elast 82:273–296CrossRefMathSciNetzbMATHGoogle Scholar
  39. Murty KG, Kabadi SN (1987) Some NP-complete problems in quadratic and nonlinear programming. Math Progr 39:117–129CrossRefMathSciNetzbMATHGoogle Scholar
  40. Ng M, Qi L, Zhou G (2009) Finding the largest eigenvalue of a nonnegative tensor. SIAM J Matrix Anal Appl 31:1090–1099CrossRefMathSciNetzbMATHGoogle Scholar
  41. Ni Q, Qi L, Wang F (2008) An eigenvalue method for testing positive definiteness of a multivariate form. IEEE Trans Autom Control 53:1096–1107CrossRefMathSciNetzbMATHGoogle Scholar
  42. Nikias CL, Mendel JM (1993) Signal processing with higher-order spectra. IEEE Signal Process Mag 10:10–37CrossRefGoogle Scholar
  43. Pearson KJ, Tan Z (2014) On spectral hypergraph theory of the adjacency tensor. Graphs Combin 30:1233–1248CrossRefMathSciNetzbMATHGoogle Scholar
  44. Qi L (2005) Eigenvalues of a real supersymmetric tensor. J Symb Comput 40:1302–1324CrossRefMathSciNetzbMATHGoogle Scholar
  45. Qi L (2013) Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl 439:228–238CrossRefMathSciNetzbMATHGoogle Scholar
  46. Qi L, Chen H, Chen Y (2018) Tensor eigenvalues and their applications. Springer, SingaporeCrossRefzbMATHGoogle Scholar
  47. Qi L, Luo Z (2017) Tensor analysis: spectral theory and special tensors, vol. 151. SIAM,Google Scholar
  48. Qi L, Shao J-Y, Wang Q (2014) Regular uniform hypergraphs, s-cycles, s-paths and their largest Laplacian \(H\)-eigenvalues. Linear Algebra Appl 443:215–227CrossRefMathSciNetzbMATHGoogle Scholar
  49. Qi L, Song Y (2014) An even order symmetric B tensor is positive definite. Linear Algebra Appl 457:303–312CrossRefMathSciNetzbMATHGoogle Scholar
  50. Qi L, Xu C, Xu Y (2014) Nonnegative tensor factorization, completely positive tensors, and a hierarchical elimination algorithm. SIAM J Matrix Anal Appl 35:1227–1241CrossRefMathSciNetzbMATHGoogle Scholar
  51. Song Y, Qi L (2014) Infinite and finite dimensional hilbert tensors. Linear Algebra Appl 451:1–14CrossRefMathSciNetzbMATHGoogle Scholar
  52. Song Y, Qi L (2015) Properties of some classes of structured tensors. J Optim Theory Appl 165:854–873CrossRefMathSciNetzbMATHGoogle Scholar
  53. Wang F, Qi L (2005) Comments on “Explicit criterion for the positive definiteness of a general quartic form”. IEEE Trans Autom Control 50:416–418CrossRefzbMATHGoogle Scholar
  54. Wang X, Che M, Qi L, Wei Y (2019) Modified gradient dynamic approach to the tensor complementarity problem. Optim Methods Softw 1–22Google Scholar
  55. Wang X, Wei Y (2015) Bounds for eigenvalues of nonsingular \(\cal{H}\)-tensor. Electron J Linear Algebra 29:3–16CrossRefMathSciNetzbMATHGoogle Scholar
  56. Wang X, Wei Y (2016) \(\cal{H}\)-tensors and nonsingular \(\cal{H}\)-tensors. Front Math China 11:557–575CrossRefMathSciNetzbMATHGoogle Scholar
  57. Wei Y, Ding W (2016) Theory and computation of tensors. Elsevier, Academic Press, AmsterdamzbMATHGoogle Scholar
  58. Xiang H, Qi L, Wei Y (2019) M-eigenvalues of the Riemann curvature tensor. Commun Math Sci 16:2301–2315CrossRefMathSciNetzbMATHGoogle Scholar
  59. Xie Z-J, Jin X-Q, Wei Y (2018) Tensor methods for solving symmetric \(\cal{M}\)-tensor systems. J Sci Comput 74:412–425CrossRefMathSciNetzbMATHGoogle Scholar
  60. Zhang L, Qi L, Zhou G (2014) M-tensors and some applications. SIAM J Matrix Anal Appl 35:437–452CrossRefMathSciNetzbMATHGoogle Scholar
  61. Zhang X, Ling C, Qi L (2012) The best rank-1 approximation of a symmetric tensor and related spherical optimization problems. SIAM J Matrix Anal Appl 33:806–821CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesFudan UniversityShanghaiPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsYunnan UniversityKunmingPeople’s Republic of China
  3. 3.School of Mathematics and StatisticsHexi UniversityZhangyePeople’s Republic of China
  4. 4.School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied MathematicsFudan UniversityShanghaiPeople’s Republic of China

Personalised recommendations