Some Bounds of Eigenvalues for Hadamard Product and Fan Product of Tensors
Tóm tắt
In this paper, some new upper bounds on the spectral radius of Hadamard product of nonnegative tensors are given. To show their sharpness, the comparisons among these bounds, including the existing one by Sun et al. (Linear Multilinear Algebra 66:1199–1214, 2018), are performed. We also present some lower bounds on the minimum eigenvalue of Fan product of irreducible strong
$${{\mathcal {M}}}$$
-tensors and their sharpness under different conditions are investigated. Some numerical examples are provided to illustrate our theoretical results.
Tài liệu tham khảo
Qi, L.Q.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)
Yang, Y.N., Yang, Q.Z.: Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J. Matrix Anal. Appl. 31, 2517–2530 (2010)
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, New York, pp. 129–132 (2005)
Zhang, L.P., Qi, L.Q., Zhou, G.L.: \({{\cal{M}}}\)-tensors and some applications. SIAM J. Matrix Anal. Appl. 35, 437–452 (2014)
Li, C.Q., Chen, Z., Li, Y.T.: A new eigenvalue inclusion set for tensors and its applications. Linear Algebra Appl. 481, 36–53 (2015)
Qi, L.Q., Sun, W.Y., Wang, Y.J.: Numerical multilinear algebra and its applications. Front. Math. China 2, 501–526 (2007)
Li, C.Q., Li, Y.T., Kong, X.: New eigenvalue inclusion sets for tensors. Numer. Linear Algebra Appl. 21, 39–50 (2014)
Li, C.Q., Wang, Y.Q., Yi, J.Y., Li, Y.T.: Bounds for the spectral radius of nonnegative tensors. J. Ind. Manag. Optim. 12, 1–16 (2016)
Li, C.Q., Zhou, J.J., Li, Y.T.: A new Brauer-type eigenvalue localization set for tensors. Linear Multilinear Algebra 64, 727–736 (2016)
Li, S.H., Li, C.Q., Li, Y.T.: A new bound for the spectral radius of nonnegative tensors. J. Inequal. Appl. 2017, 88 (2017)
Zhao, J.X., Sang, C.L.: Two new lower bounds for the minimum eigenvalue of \({{\cal{M}}}\)-tensors. J. Inequal. Appl. 2016, 268 (2016)
Zhao, J.X., Sang, C.L.: An eigenvalue localization set for tensors and its applications. J. Inequal. Appl. 2017, 59 (2017)
He, J., Huang, T.Z.: Inequalities for \({{\cal{M}}}\)-tensors. J. Inequal. Appl. 2014, 114 (2014)
Qi, L.Q.: Hankel tensors: Associated Hankel matrices and Vandermonde decomposition. Commun. Math. Sci. 13, 113–125 (2015)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1985)
Qi, L.Q., Xu, C.Q., Xu, Y.: Nonnegative tensor factorization, completely positive tensors, and a hierarchical elimination algorithm. SIAM J. Matrix Anal. Appl. 35, 1227–1241 (2014)
Kannan, M.R., Shaked-Monderer, N., Berman, A.: Some properties of strong \({{\cal{H}}}\)-tensors and general \({{\cal{H}}}\)-tensors. Linear Algebra Appl. 476, 42–55 (2015)
Zhou, J., Sun, L.Z., Wei, Y.P., Bu, C.J.: Some characterizations of \({{\cal{M}}}\)-tensors via digraphs. Linear Algebra Appl. 495, 190–198 (2016)
Xu, Y.Y., Li, Y.T., Li, Z.B.: Some results on the Hadamard product of tensors. Bull. Iran. Math. Sci. Soc. (2018). https://doi.org/10.1007/s41980-018-00193-0
Li, Y.T., Li, Y.Y., Wang, R.W., Wang, Y.Q.: Some new bounds on eigenvalues of the Hadamard product and the Fan product of matrices. Linear Algebra Appl. 432, 536–545 (2010)
Liu, Q.B., Chen, G.L.: On two inequalities for the Hadamard product and the Fan product of matrices. Linear Algebra Appl. 431, 974–984 (2009)
Zhao, L.L., Liu, Q.B.: Some inequalities on the spectral radius of matrices. J. Inequal. Appl. 5, 1–12 (2018)
Fang, M.Z.: Bounds on the eigenvalues of the Hadamard product and the Fan product of matrices. Linear Algebra Appl. 425, 7–15 (2007)
Huang, R.: Some inequalities for the Hadamard product and the Fan product of matrices. Linear Algebra Appl. 428, 1551–1559 (2008)
Zhou, D.M., Chen, G.L., Wu, G.X., Zhang, X.Y.: On some new bounds for eigenvalues of the Hadamard product and the Fan product of matrices. Linear Algebra Appl. 438, 1415–1426 (2013)
Sun, L.Z., Zheng, B.D., Zhou, J., Yan, H.: Some inequalities for the Hadamard product of tensors. Linear Multilinear Algebra. 66, 1199–1214 (2018)
Ding, W.Y., Qi, L.Q.: \({{\cal{M}}}\)-tensors and nonsingular \({{\cal{M}}}\)-tensors. Linear Algebra Appl. 439, 3264–3278 (2013)
Cheng, G.H.: New bounds for the minimum eigenvalue of the Fan product of two \(M\)-matrices. Czech. Math. J. 64, 63–68 (2014)
Chang, K.C., Pearson, K., Zhang, T.: Perron–Frobenius theorem for nonnegative tensors. Commun. Math. Sci. 6, 507–520 (2008)
Hardy, G.H., Littlewood, J.E., Polya, G.: Inequality. Cambridge University Press, Cambridge (1934)
Zheng, Y.M., Cui, R.Q.: Upper bound of the spectral radius for Hadamard product of nonnegative matrices. J. Henan Polytech. Univ.(Nat. Sci.) 29, 543–546 (2010)