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Upper bounds for Z$ _1 $-eigenvalues of generalized Hilbert tensors
School of Mathematics and Information Science, Henan Normal University, XinXiang HeNan, China |
The Z$ _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.
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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