American Institute of Mathematical Sciences

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March  2020, 16(2): 911-918. doi: 10.3934/jimo.2018184

Upper bounds for Z$ _1 $-eigenvalues of generalized Hilbert tensors

Juan Meng 1, and  Yisheng Song 1,,

School of Mathematics and Information Science, Henan Normal University, XinXiang HeNan, China

* Corresponding author: Yisheng Song

Received  March 2018 Revised  June 2018 Published  March 2020 Early access  December 2018

Fund Project: The first author is supported by the National Natural Science Foundation of China (Grant No.11571095, 11601134, 11701154)

The Z$ _1 $-eigenvalue of tensors (hypermatrices) was widely used to discuss the properties of higher order Markov chains and transition probability tensors. In this paper, we extend the concept of Z$ _1 $-eigenvalue from finite-dimensional tensors to infinite-dimensional tensors, and discuss the upper bound of such eigenvalues for infinite-dimensional generalized Hilbert tensors. Furthermore, an upper bound of Z$ _1 $-eigenvalue for finite-dimensional generalized Hilbert tensor is obtained also.

Keywords: Generalized Hilbert tensor, $ Z_1 $-eigenvalue, spectral radius, Hilbert inequalities.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Juan Meng, Yisheng Song. Upper bounds for Z$ _1 $-eigenvalues of generalized Hilbert tensors. Journal of Industrial and Management Optimization, 2020, 16 (2) : 911-918. doi: 10.3934/jimo.2018184
References:
[1]

A. Aleman, A. Montes-Rodriguez and A. Sarafoleanu, The eigenfunctions of the Hilbert matrix, Constr Approx., 36 (2012), 353-374.  doi: 10.1007/s00365-012-9157-z.

[2]

K. C. Chang, K. Pearson and T. Zhang, On the uniqueness and non-uniqueness of the positive Z-eigenvector for transition probability tensors, J. Math. Anal. Appl., 408 (2013), 525-540.  doi: 10.1016/j.jmaa.2013.04.019.

[3]

K. C. Chang, K. Pearson and T. Zhang, Some variational principles for $Z$-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438 (2013), 4166-4182.  doi: 10.1016/j.laa.2013.02.013.

[4]

K. C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.  doi: 10.4310/CMS.2008.v6.n2.a12.

[5]

H. Chen and L. Qi, Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors, J. Ind. Manag. Optim., 11 (2015), 1263-1274.  doi: 10.3934/jimo.2015.11.1263.

[6]

H. Chen and Y. Wang, On computing minimal H-eigenvalue of sign-structured tensors, Front. Math. China, 12 (2017), 1289-1302.  doi: 10.1007/s11464-017-0645-0.

[7]

H. Chen, L. Qi and Y. Song, Column sufficient tensors and tensor complementarity problems, Front. Math. China, 13 (2018), 255-276.  doi: 10.1007/s11464-018-0681-4.

[8]

M. D. Choi, Tricks for treats with the hilbert matrix, Amer. Math. Monthly, 90 (1983), 301-312.  doi: 10.1080/00029890.1983.11971218.

[9]

J. Culp, K. Pearson and T. Zhang, On the uniqueness of the $Z_1$-eigenvector of transition probability tensors, Linear Multilinear Algebra, 65 (2017), 891-896.  doi: 10.1080/03081087.2016.1211130.

[10]

H. Frazer, Note on Hilbert's Inequality, J. London Math. Soc., 21 (1946), 7-9.  doi: 10.1112/jlms/s1-21.1.7.

[11]

J. He and T. Huang, Upper bound for the largest $Z$-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.

[12]

J. He, Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl., 20 (2016), 1290-1301. 

[13]

J. He, Y. M. Liu, H. Ke, J. K. Tian and X. Li, Bounds for the $Z$-spectral radius of nonnegative tensors, SpringerPlus, 5 (2016), 1727.

[14]

D. Hilbert, Ein beitrag zur theorie des legendre'schen polynoms, Acta Mathematica, 18 (1894), 155-159.  doi: 10.1007/BF02418278.

[15]

C. K. Hill, On the singly-infinite Hilbert matrix, J. London Math. Soc., 35 (1960), 17-29.  doi: 10.1112/jlms/s1-35.1.17.

[16]

A. E. Ingham, A Note on Hilbert's Inequality, J. London Math. Soc., 11 (1936), 237-240.  doi: 10.1112/jlms/s1-11.3.237.

[17]

T. Kato, On the hilbert matrix, Proc. American Math. Soc., 8 (1957), 73-81.  doi: 10.1090/S0002-9939-1957-0083965-4.

[18]

W. Li, D. Liu and S. W. Vong, $Z$-eigenpair bounds for an irreducible nonnegative tensor, Linear Algebra Appl., 483 (2015), 182-199.  doi: 10.1016/j.laa.2015.05.033.

[19]

L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, 1st IEEE International workshop on computational advances of multi-tensor adaptive processing, 13/15 (2005), 129-132. 

[20]

W. Magnus, On the spectrum of Hilbert's matrix, American J. Math., 72 (1950), 699-704.  doi: 10.2307/2372284.

[21]

W. Mei and Y. Song, Infinite and finite dimensional generalized Hilbert tensors, Linear Algebra Appl., 532 (2017), 8-24.  doi: 10.1016/j.laa.2017.05.052.

[22]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.

[23]

L. Qi, Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines, J. Symbolic Comput., 41 (2006), 1309-1327.  doi: 10.1016/j.jsc.2006.02.011.

[24]

L. Qi, Symmetric nonnegative tensors and copositive tensors, Linear Algebra Appl., 439 (2013), 228-238.  doi: 10.1016/j.laa.2013.03.015.

[25]

L. Qi, Hankel tensors: Associated Hankel matrices and Vandermonde decomposition, Commun. Math. Sci., 13 (2015), 113-125.  doi: 10.4310/CMS.2015.v13.n1.a6.

[26]

M. Rosenblum, On the Hilbert matrix Ⅰ, Proc.Am. Math. Soc., 9 (1958), 137-140.  doi: 10.2307/2033411.

[27]

Y. Song and L. Qi, Infinite and finite dimensional Hilbert tensors, Linear Algebra Appl., 451 (2014), 1-14.  doi: 10.1016/j.laa.2014.03.023.

[28]

Y. Song and L. Qi, Infinite dimensional Hilbert tensors on spaces of analytic functions, Commun. Math. Sci., 15 (2017), 1897-1911.  doi: 10.4310/CMS.2017.v15.n7.a5.

[29]

Y. Song and L. Qi, Positive eigenvalue-eigenvector of nonlinear positive mappings, Front. Math. China, 9 (2014), 181-199.  doi: 10.1007/s11464-013-0258-1.

[30]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM. J. Matrix Anal. Appl., 34 (2013), 1581-1595.  doi: 10.1137/130909135.

[31]

Y. Song and L. Qi, Necessary and sufficient conditions for copositive tensors, Linear Multilinear Algebra, 63 (2015), 120-131.  doi: 10.1080/03081087.2013.851198.

[32]

O. Taussky, A remark concerning the characteristic roots of the finite segments of the Hilbert matrix, Quarterly J. Math. Oxford ser., 20 (1949), 80-83.  doi: 10.1093/qmath/os-20.1.80.

[33]

Y. Wang, K. Zhang and H. Sun, Criteria for strong H-tensors, Front. Math. China, 11 (2016), 577-592.  doi: 10.1007/s11464-016-0525-z.

[34]

Y. Wang, G. Zhou and L. Caccetta, Nonsingular H-tensors and its criteria, J. Industr. Manag. Optim., 12 (2016), 1173-1186.  doi: 10.3934/jimo.2016.12.1173.

[35]

Y. Wang; L. Caccetta and G. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numerical Linear Algebra with Applications, 22 (2015), 1059-1076.  doi: 10.1002/nla.1996.

[36]

Y. Wang, L. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numerical Linear Algebra with Applications, 16 (2009), 589-601.  doi: 10.1002/nla.633.

[37]

C. Xu, Hankel tensors, Vandermonde tensors and their positivities, Linear Algebra Appl., 491 (2016), 56-72.  doi: 10.1016/j.laa.2015.02.012.

[38]

Q. Yang and Y. Yang, Further Results for Perron-Frobenius Theorem for Nonnegative Tensors Ⅱ, SIAM. J. Matrix Anal. Appl., 32 (2011), 1236-1250.  doi: 10.1137/100813671.

[39]

Q. Yang and Y. Yang, Further Results for Perron-Frobenius Theorem for Nonnegative Tensors, SIAM. J. Matrix Anal. Appl., 31 (2010), 2517-2530.  doi: 10.1137/090778766.

[40]

K. Zhang and Y. Wang, An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, J. Compu. Appl. Math., 305 (2016), 1-10.  doi: 10.1016/j.cam.2016.03.025.

show all references

References:
[1]

A. Aleman, A. Montes-Rodriguez and A. Sarafoleanu, The eigenfunctions of the Hilbert matrix, Constr Approx., 36 (2012), 353-374.  doi: 10.1007/s00365-012-9157-z.

[2]

K. C. Chang, K. Pearson and T. Zhang, On the uniqueness and non-uniqueness of the positive Z-eigenvector for transition probability tensors, J. Math. Anal. Appl., 408 (2013), 525-540.  doi: 10.1016/j.jmaa.2013.04.019.

[3]

K. C. Chang, K. Pearson and T. Zhang, Some variational principles for $Z$-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438 (2013), 4166-4182.  doi: 10.1016/j.laa.2013.02.013.

[4]

K. C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.  doi: 10.4310/CMS.2008.v6.n2.a12.

[5]

H. Chen and L. Qi, Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors, J. Ind. Manag. Optim., 11 (2015), 1263-1274.  doi: 10.3934/jimo.2015.11.1263.

[6]

H. Chen and Y. Wang, On computing minimal H-eigenvalue of sign-structured tensors, Front. Math. China, 12 (2017), 1289-1302.  doi: 10.1007/s11464-017-0645-0.

[7]

H. Chen, L. Qi and Y. Song, Column sufficient tensors and tensor complementarity problems, Front. Math. China, 13 (2018), 255-276.  doi: 10.1007/s11464-018-0681-4.

[8]

M. D. Choi, Tricks for treats with the hilbert matrix, Amer. Math. Monthly, 90 (1983), 301-312.  doi: 10.1080/00029890.1983.11971218.

[9]

J. Culp, K. Pearson and T. Zhang, On the uniqueness of the $Z_1$-eigenvector of transition probability tensors, Linear Multilinear Algebra, 65 (2017), 891-896.  doi: 10.1080/03081087.2016.1211130.

[10]

H. Frazer, Note on Hilbert's Inequality, J. London Math. Soc., 21 (1946), 7-9.  doi: 10.1112/jlms/s1-21.1.7.

[11]

J. He and T. Huang, Upper bound for the largest $Z$-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.

[12]

J. He, Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl., 20 (2016), 1290-1301. 

[13]

J. He, Y. M. Liu, H. Ke, J. K. Tian and X. Li, Bounds for the $Z$-spectral radius of nonnegative tensors, SpringerPlus, 5 (2016), 1727.

[14]

D. Hilbert, Ein beitrag zur theorie des legendre'schen polynoms, Acta Mathematica, 18 (1894), 155-159.  doi: 10.1007/BF02418278.

[15]

C. K. Hill, On the singly-infinite Hilbert matrix, J. London Math. Soc., 35 (1960), 17-29.  doi: 10.1112/jlms/s1-35.1.17.

[16]

A. E. Ingham, A Note on Hilbert's Inequality, J. London Math. Soc., 11 (1936), 237-240.  doi: 10.1112/jlms/s1-11.3.237.

[17]

T. Kato, On the hilbert matrix, Proc. American Math. Soc., 8 (1957), 73-81.  doi: 10.1090/S0002-9939-1957-0083965-4.

[18]

W. Li, D. Liu and S. W. Vong, $Z$-eigenpair bounds for an irreducible nonnegative tensor, Linear Algebra Appl., 483 (2015), 182-199.  doi: 10.1016/j.laa.2015.05.033.

[19]

L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, 1st IEEE International workshop on computational advances of multi-tensor adaptive processing, 13/15 (2005), 129-132. 

[20]

W. Magnus, On the spectrum of Hilbert's matrix, American J. Math., 72 (1950), 699-704.  doi: 10.2307/2372284.

[21]

W. Mei and Y. Song, Infinite and finite dimensional generalized Hilbert tensors, Linear Algebra Appl., 532 (2017), 8-24.  doi: 10.1016/j.laa.2017.05.052.

[22]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.

[23]

L. Qi, Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines, J. Symbolic Comput., 41 (2006), 1309-1327.  doi: 10.1016/j.jsc.2006.02.011.

[24]

L. Qi, Symmetric nonnegative tensors and copositive tensors, Linear Algebra Appl., 439 (2013), 228-238.  doi: 10.1016/j.laa.2013.03.015.

[25]

L. Qi, Hankel tensors: Associated Hankel matrices and Vandermonde decomposition, Commun. Math. Sci., 13 (2015), 113-125.  doi: 10.4310/CMS.2015.v13.n1.a6.

[26]

M. Rosenblum, On the Hilbert matrix Ⅰ, Proc.Am. Math. Soc., 9 (1958), 137-140.  doi: 10.2307/2033411.

[27]

Y. Song and L. Qi, Infinite and finite dimensional Hilbert tensors, Linear Algebra Appl., 451 (2014), 1-14.  doi: 10.1016/j.laa.2014.03.023.

[28]

Y. Song and L. Qi, Infinite dimensional Hilbert tensors on spaces of analytic functions, Commun. Math. Sci., 15 (2017), 1897-1911.  doi: 10.4310/CMS.2017.v15.n7.a5.

[29]

Y. Song and L. Qi, Positive eigenvalue-eigenvector of nonlinear positive mappings, Front. Math. China, 9 (2014), 181-199.  doi: 10.1007/s11464-013-0258-1.

[30]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM. J. Matrix Anal. Appl., 34 (2013), 1581-1595.  doi: 10.1137/130909135.

[31]

Y. Song and L. Qi, Necessary and sufficient conditions for copositive tensors, Linear Multilinear Algebra, 63 (2015), 120-131.  doi: 10.1080/03081087.2013.851198.

[32]

O. Taussky, A remark concerning the characteristic roots of the finite segments of the Hilbert matrix, Quarterly J. Math. Oxford ser., 20 (1949), 80-83.  doi: 10.1093/qmath/os-20.1.80.

[33]

Y. Wang, K. Zhang and H. Sun, Criteria for strong H-tensors, Front. Math. China, 11 (2016), 577-592.  doi: 10.1007/s11464-016-0525-z.

[34]

Y. Wang, G. Zhou and L. Caccetta, Nonsingular H-tensors and its criteria, J. Industr. Manag. Optim., 12 (2016), 1173-1186.  doi: 10.3934/jimo.2016.12.1173.

[35]

Y. Wang; L. Caccetta and G. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numerical Linear Algebra with Applications, 22 (2015), 1059-1076.  doi: 10.1002/nla.1996.

[36]

Y. Wang, L. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numerical Linear Algebra with Applications, 16 (2009), 589-601.  doi: 10.1002/nla.633.

[37]

C. Xu, Hankel tensors, Vandermonde tensors and their positivities, Linear Algebra Appl., 491 (2016), 56-72.  doi: 10.1016/j.laa.2015.02.012.

[38]

Q. Yang and Y. Yang, Further Results for Perron-Frobenius Theorem for Nonnegative Tensors Ⅱ, SIAM. J. Matrix Anal. Appl., 32 (2011), 1236-1250.  doi: 10.1137/100813671.

[39]

Q. Yang and Y. Yang, Further Results for Perron-Frobenius Theorem for Nonnegative Tensors, SIAM. J. Matrix Anal. Appl., 31 (2010), 2517-2530.  doi: 10.1137/090778766.

[40]

K. Zhang and Y. Wang, An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, J. Compu. Appl. Math., 305 (2016), 1-10.  doi: 10.1016/j.cam.2016.03.025.

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