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Brauer-type bounds for Hadamard product of nonnegative tensors

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Abstract

In this paper, we establish some Brauer-type bounds for the spectral radius of Hadamard product of two nonnegative tensors based on Brauer-type inclusion set, which are shown to be sharper than the existing bounds established in the literature. The validity of the obtained results is theoretically and numerically tested.

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References

  1. Bloy L, Verma R. On computing the underlying fiber directions from the diffusion orientation distribution function. In: Metaxas D, Axel L, Fichtinger G, Székely G, eds. Medical Image Computing and Computer-Assisted Intervention—MICCAI 2008, Part I. Lecture Notes in Comput Sci, Vol 5241. Berlin: Springer, 2008, 1–8

    Google Scholar 

  2. Bu C, Jin X, Li H, Deng C. Brauer-type eigenvalue inclusion sets and the spectral radius of tensors. Linear Algebra Appl, 2017, 512: 234–248

    Article  MathSciNet  Google Scholar 

  3. Che M, Wei Y. Theory and Computation of Complex Tensors and its Applications. Singapore: Springer, 2020

    Book  Google Scholar 

  4. Chen H, Qi L, Song Y. Column sufficient tensors and tensor complementarity problems. Front Math China, 2018, 13: 255–276

    Article  MathSciNet  Google Scholar 

  5. Ding W, Wei Y. Solving multi-linear systems with M-tensors. J Sci Comput, 2016, 68: 689–715

    Article  MathSciNet  Google Scholar 

  6. Fang F. Bounds on eigenvalues of Hadamard product and the Fan product of matrices. Linear Algebra Appl, 2007, 425: 7–15

    Article  MathSciNet  Google Scholar 

  7. Friedland S, Gaubert S, Han L. Perron-Frobenius theorem for nonnegative multilinear forms and extensions. Linear Algebra Appl, 2013, 438: 738–749

    Article  MathSciNet  Google Scholar 

  8. Gao L, Cao Z, Wang G. Input-to-state stability and integral input-to-state stability for impulsive switched systems with time-delay under asynchronous switching. Nonlinear Anal Hybrid Syst, 2019, 34: 248–263

    Article  MathSciNet  Google Scholar 

  9. Gao L, Luo F, Yan Z. Finite-time annular domain stability of impulsive switched systems: mode-dependent parameter approach. Internat J Control, 2019, 92: 1381–1392

    Article  MathSciNet  Google Scholar 

  10. Horn R, Johnson C. Topics in Matrix Analysis. Cambridge: Cambridge Univ Press, 1985

    Book  Google Scholar 

  11. Hu S, Huang Z, Ling C, Qi L. On determinants and eigenvalue theory of tensors. J Symbolic Comput, 2013, 50: 508–531

    Article  MathSciNet  Google Scholar 

  12. Hu S, Huang Z, Qi L. Strictly nonnegative tensors and nonnegative tensor partition. Sci China Math, 2014, 57: 181–195

    Article  MathSciNet  Google Scholar 

  13. Huang R. Some inequalities for the Hadamard product and the Fan product of matrices. Linear Algebra Appl, 2008, 428: 1551–1559

    Article  MathSciNet  Google Scholar 

  14. Jutten C, Herault J. Blind separation of sources, part I: an adaptive algorithm based on neuromimetic architecture. Signal Processing, 1991, 24(1): 1–10

    Article  Google Scholar 

  15. Kolda T, Bader B. Tensor decompositions and applications. SIAM Review, 2009, 51: 455–500

    Article  MathSciNet  Google Scholar 

  16. Li C, Li Y, Kong X. New eigenvalue inclusion sets for tensors. Numer Linear Algebra Appl, 2014, 21: 39–50

    Article  MathSciNet  Google Scholar 

  17. Li Y, Chen F, Wang D. New lower bounds on eigenvalue of the Hadamard product of an M-matrix and its inverse. Linear Algebra Appl, 2009, 430: 1423–1431

    Article  MathSciNet  Google Scholar 

  18. Lim L H. Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, Puerto Vallarta, 2005. 2005, 129–132

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302–1324

    Article  MathSciNet  Google Scholar 

  21. Qi L. Hankel tensors: associated Hankel matrices and Vandermonde decomposition. Commun Math Sci, 2015, 13: 113–125

    Article  MathSciNet  Google Scholar 

  22. Qi L, Luo Z. Tensor Analysis: Spectral Theory and Special Tensors. Philadelphia: SIAM, 2017

    Book  Google Scholar 

  23. Sun L, Zheng B, Zhou J, Yan H. Some inequalities for the Hadamard product of tensors. Linear Multilinear Algebra, 2018, 66: 1199–1214

    Article  MathSciNet  Google Scholar 

  24. Wang G, Wang Y, Liu L. Bound estimations on the eigenvalues for Fan product of M-tensors. Taiwanese J Math, 2019, 23: 751–766

    Article  MathSciNet  Google Scholar 

  25. Wang G, Wang Y, Zhang Y. Some inequalities for the Fan product of M-tensors. J Inequal Appl, 2018, 2018: 257

    Article  MathSciNet  Google Scholar 

  26. Wang G, Wang Y, Zhang Y. Brauer-type upper bounds for Z-Spectral radius of weakly symmetric nonnegative tensors. J Math Inequal, 2019, 13(4): 1105–1116

    Article  MathSciNet  Google Scholar 

  27. Wang G, Zhou G, Caccetta L. Z-eigenvalue inclusion theorems for tensors. Discrete Contin Dyn Syst Ser B, 2017, 22: 187–198

    MathSciNet  MATH  Google Scholar 

  28. Wang G, Zhou G, Caccetta L. Sharp Brauer-type eigenvalue inclusion theorems for tensors. Pac J Optim, 2018, 14: 227–244

    MathSciNet  Google Scholar 

  29. Wang X, Chen H, Wang Y. Solution structures of tensor complementarity problem. Front Math China, 2018, 13: 935–945

    Article  MathSciNet  Google Scholar 

  30. Wang Y, Zhang K, Sun H. Criteria for strong H-tensors. Front Math China, 2016, 11: 577–592

    Article  MathSciNet  Google Scholar 

  31. Wang Y, Zhou G, Caccetta L. Convergence analysis of a block improvement method for polynomial optimization over unit spheres. Numer Linear Algebra Appl, 2015, 22: 1059–1076

    Article  MathSciNet  Google Scholar 

  32. Wang Y, Zhou G, Caccetta L. Nonsingular H-tensor and its criteria. J Ind Manag Optim, 2016, 12: 1173–1186

    Article  MathSciNet  Google Scholar 

  33. Yang Q, Yang Y. Further results for Perron-Frobenius theorem for nonnegative tensors II. SIAM J Matrix Anal Appl, 2011, 32: 1236–1250

    Article  MathSciNet  Google Scholar 

  34. Zhou D, Chen G, Wu G, Zhang X. On some new bounds for eigenvalues of the Hadamard product and the Fan product of matrices. Linear Algebra Appl, 2013, 438: 1415–1426

    Article  MathSciNet  Google Scholar 

  35. Zhou G, Qi L, Wu S. Efficient algorithms for computing the largest eigenvalue of a nonnegative tensor. Front Math China, 2013, 8: 155–168

    Article  MathSciNet  Google Scholar 

  36. Zhou G, Wang G, Qi L, Alqahtani M. A fast algorithm for the spectral radii of weakly reducible nonnegative tensors. Numer Linear Algebra Appl, 2018, 25(2): e2134

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11671228).

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Correspondence to Yiju Wang.

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Wang, G., Zhang, Y. & Wang, Y. Brauer-type bounds for Hadamard product of nonnegative tensors. Front. Math. China 15, 555–570 (2020). https://doi.org/10.1007/s11464-020-0840-2

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  • DOI: https://doi.org/10.1007/s11464-020-0840-2

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