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Some criteria for identifying \({{\mathcal {H}}}\)-tensors and its applications

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Abstract

We present some new criteria for \({{\mathcal {H}}}\)-tensors by constructing different positive diagonal matrices and applying some techniques of inequalities. As an application, some sufficient conditions for the positive definiteness of an even-order real symmetric tensor are derived. Some numerical examples are presented to illustrate the theoretical results.

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References

  1. Qi, L.Q.: Eigenvalues of a real supersymetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)

    Article  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, Puerto Vallarta, pp. 129–132 (2005)

  3. Yang, Y.N., Yang, Q.Z.: Fruther results for Perron–Frobenius theorem for nonnegative tensors. SIAM. J. Matrix Anal. Appl. 31, 2517–2530 (2010)

    Article  MathSciNet  Google Scholar 

  4. Ding, W.Y., Qi, L.Q., Wei, Y.M.: \(M\)-tensors and nonsingular \(M\)-tensors. Linear Algebra Appl. 439, 3264–3278 (2013)

    Article  MathSciNet  Google Scholar 

  5. Kannana, M.R., Mondererb, N.S., Bermana, A.: Some properties of strong \(\cal{H}\)-tensors and general \(\cal{H}\)-tensors. Linear Algebra Appl. 476, 42–55 (2015)

    Article  MathSciNet  Google Scholar 

  6. Wang, F., Sun, D.S., Zhao, J.X., Li, C.Q.: New practical criteria for \(\cal{H}\)-tensors and its application. Linear Multilinear Algebra 65(2), 269–283 (2017)

    Article  MathSciNet  Google Scholar 

  7. Zhang, L.P., Qi, L.Q., Zhou, G.L.: \(M\)-tensors and some applications. SIAM J. Matrix Anal. Appl. 32, 437–452 (2014)

    Article  MathSciNet  Google Scholar 

  8. Qi, L.Q., Yu, G.H., Xu, Y.: Nonnegative diffision orientation distribution function. J. Math. Imaging Vision 45, 103–113 (2013)

    Article  MathSciNet  Google Scholar 

  9. Hu, S.L., Qi, L.Q., Shao, J.Y.: Cored hypergraphs, power hypergraphs and their Laplacian eigenvalues. Linear Algebra Appl. 439, 2980–2998 (2013)

    Article  MathSciNet  Google Scholar 

  10. 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)

    Article  MathSciNet  Google Scholar 

  11. Ni, Q., Qi, L.Q., Wang, F.: An eigenvalue method for testing positive definiteness of a multivariate form. IEEE Trans. Automat. Control 53(3), 1096–1107 (2008)

    Article  MathSciNet  Google Scholar 

  12. Song, Y.S., Qi, L.Q.: Necessary and sufficient conditions for copositive tensors. Linear Multilinear Algebra 63(1), 120–131 (2015)

    Article  MathSciNet  Google Scholar 

  13. Zhang, K.L., Wang, Y.J.: An \(\cal{H}\)-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms. J. Comput. Appl. Math. 305, 1–10 (2016)

    Article  MathSciNet  Google Scholar 

  14. Bose, N.K., Modaress, A.R.: General procedure for multivariable polynomial positivity with control applications. IEEE Trans. Automat. Control 21, 596–601 (1976)

    Article  MathSciNet  Google Scholar 

  15. Li, C.Q., Wang, F., Zhao, J.X., Zhu, Y., Li, Y.T.: Criterions for the positive definiteness of real supersymmetric tensors. J. Comput. Appl. Math. 255, 1–14 (2014)

    Article  MathSciNet  Google Scholar 

  16. Li, Y.T., Liu, Q.L., Qi, L.Q.: Programmable criteria for strong \(\cal{H}\)-tensors. Numer. Algorithms 74, 199–211 (2017)

    Article  MathSciNet  Google Scholar 

  17. Wang, Y., Zhou, G.L., Caccetta, L.: Nonsingular \(\cal{H}\)-tensor and its criteria. J. Ind. Manag. Optim. 12, 1173–1186 (2016)

    Article  MathSciNet  Google Scholar 

  18. Wang, Y.J., Zhang, K.L., Sun, H.C.: Criteria for strong \(\cal{H}\)-tensors. Front. Math. China 11(3), 577–592 (2016)

    Article  MathSciNet  Google Scholar 

  19. Li, Y.T., Liu, Q.L., Qi, L.Q.: Criteria for strong \(\cal{H}\)-tensor. Front. Math. China 11, 577–592 (2016)

    Article  MathSciNet  Google Scholar 

  20. Qi, L.Q., Song, Y.S.: An even order symmetric \(B\)-tensor is positive definite. Linear Algebra Appl. 457, 303–312 (2014)

    Article  MathSciNet  Google Scholar 

  21. Li, C.Q., Li, Y.T.: Double \(B\)-tensors and quasi-double \(B\)-tensors. Linear Algebra Appl. 466, 343–356 (2015)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors wish to give their sincere thanks to the anonymous referees for their valuable suggestions and helpful comments, which help improve the quality of the paper significantly. This work was supported by the National Natural Science Foundation of China (11601473), the Foundation of Science and Technology Department of Guizhou Province (20191161, 20181079), the Natural Science Programs of Education Department of Guizhou Province (2015420), and the Research Foundation of Guizhou Minzu University (2017YB068).

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  1. College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, 550025, Guizhou, People’s Republic of China

    Feng Wang, Deshu Sun & Yumei Xu

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  1. Feng Wang
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  2. Deshu Sun
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  3. Yumei Xu
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Correspondence to Feng Wang.

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Wang, F., Sun, D. & Xu, Y. Some criteria for identifying \({{\mathcal {H}}}\)-tensors and its applications. Calcolo 56, 19 (2019). https://doi.org/10.1007/s10092-019-0316-5

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  • DOI: https://doi.org/10.1007/s10092-019-0316-5

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