Upper bounds for eigenvalues of Cauchy-Hankel tensors

Abstract

We present upper bounds of eigenvalues for finite and infinite dimensional Cauchy-Hankel tensors. It is proved that an m-order infinite dimensional Cauchy-Hankel tensor defines a bounded and positively (m－1)-homogeneous operator from l¹ into l^p (1<p<∞); and two upper bounds of corresponding positively homogeneous operator norms are given. Moreover, for a fourth-order real partially symmetric Cauchy-Hankel tensor, suffcient and necessary conditions of M-positive definiteness are obtained, and an upper bound of M-eigenvalue is also shown.

Keywords

Cauchy-Hankel tensor / eigenvalues / upper bound / M-positive definite

References

[1] Chang K C, Zhang T. On the uniqueness and non-uniqueness of the positive Z-eigenvector for transition probability tensors. J Math Anal App, 2013, 408: 525–540

[2] Che H, Chen H, Wang Y. M-positive semi-definiteness and M-positive definiteness of fourth-order partially symmetric Cauchy tensors. J Inequal Appl, 2019, 2019: 32

[3] Che H, Chen H, Wang Y. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. J Ind Manag Optim, 2020, 16(1): 309–324

[4] Chen H, Li G, Qi L. Further results on Cauchy tensors and Hankel tensors. Appl Math Comput, 2016, 275: 50–62

[5] Chen H, Qi L. Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors. J Ind Manag Optim, 2015, 11(4): 1263–1274

[6] Culp J, Pearson K, Zhang T. On the uniqueness of the Z₁-eigenvector of transition probability tensors. Linear Multilinear Algebra, 2017, 65: 891–896

[7] Frazer H. Note on Hilbert's inequality. J Lond Math Soc, 1946, 21: 7–9

[8] Güngör A D. Lower bounds for the norms of Cauchy-Toeplitz and Cauchy-Hankel matrices. Appl Math Comput, 2004, 157(3): 599–604

[9] Hardy G H, Littlewood J E, Pólya G. Inequalities. Cambridge: Cambridge Univ Press, 1952

[10] He J, Xu G, Liu Y. Some inequalities for the minimum M-eigenvalue of elasticity M-tensors. J Ind Manag Optim, 2020, 16(6): 3035–3045

[11] Kuang J, Debnath L. On new generalizations of Hilbert's inequality and their applications. J Math Anal Appl, 2000, 245: 248–265

[12] Li S, Li Y. Checkable criteria for the M-positive definiteness of fourth-order partially symmetric tensors. Bull Iranian Math Soc, 2020, 46: 1455–1463

[13] Lim L H. Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the 1st IEEE International Workshop on Computational Advances in Multi-Tensor Adaptive Processing, Vol 1, 2005. 2005, 129–132

[14] Mei W, Song Y. Infinite and finite dimensional generalized Hilbert tensors. Linear Algebra Appl, 2017, 532: 8–24

[15] Meng J, Song Y. Upper bounds for Z₁-eigenvalues of generalized Hilbert tensors. J Ind Manag Optim, 2020, 16(2): 911–918

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

[17] Qi L. Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines. J Symbolic Comput, 2006, 41: 1309–1327

[18] Qi L, Dai H, Han D. Conditions for strong ellipticity and M-eigenvalues. Front Math China, 2009, 4: 349–364

[19] Solak S, Bozkurt D. Some bounds on l_p matrix and l_p operator norms of almost circulant, Cauchy-Toeplitz and Cauchy-Hankel matrices. Math Comput Appl, 2002, 7(3): 211–218

[20] Solak S, Bozkurt D. On the spectral norms of Cauchy-Toeplitz and Cauchy-Hankel matrices. Appl Math Comput, 2003, 140(2): 231–238

[21] Song Y, Mei W. Structural properties of tensors and complementarity problems. J Optim Theory Appl, 2018, 176: 289–305

[22] Song Y, Qi L. Infinite and finite dimensional Hilbert tensors. Linear Algebra Appl, 2014, 451: 1–14

[23] Türkmen R, Bozkurt D. On the bounds for the norms of Cauchy-Toeplitz and Cauchy- Hankel matrices. Appl Math Comput, 2002, 132: 633–642

[24] Wang G, Sun L, Liu L. M-eigenvalues-based sufficient conditions for the positive definiteness of fourth-order partially symmetric tensors. Complexity, 2020, (3): 1–8

[25] Zhang Y, Sun L, Wang G. Sharp bounds on the minimum M-eigenvalue of elasticity M-tensors. Mathematics, 2020, 8(2): 250

RIGHTS & PERMISSIONS

Higher Education Press