Brauer-type bounds for Hadamard product of nonnegative tensors

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.

Keywords

Hadamard product / nonnegative tensor / Brauer-type inclusion set / spectral radius

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

[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

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

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

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

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

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

[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

[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

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

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

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

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

[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

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

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

[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

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

[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

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

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

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

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

[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

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

[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

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

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

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

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

[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

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

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

[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

[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

[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

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