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Brualdi-type inequalities on the minimum eigenvalue for the Fan product of M-tensors
School of Management Science, Qufu Normal University, Rizhao, Shandong 276826, China |
In this paper, we focus on some inequalities for the Fan product of $ M $-tensors. Based on Brualdi-type eigenvalue inclusion sets of $ M $-tensors and similarity transformation methods, we establish Brualdi-type inequalities on the minimum eigenvalue for the Fan product of two $ M $-tensors. Furthermore, we discuss the advantages of different Brualdi-type inequalities. Numerical examples verify the validity of the conclusions.
References:
[1] |
L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, in Medical Image Computing and Computer-Assisted Intervention, Springer, 2008, 1-8.
doi: 10.1007/978-3-540-85988-8_1. |
[2] |
C. Bu, Y. Wei, L. Sun and J. Zhou,
Brualdi-type eigenvalue inclusion sets of tensors, Linear Algebra Appl., 480 (2015), 168-175.
doi: 10.1016/j.laa.2015.04.034. |
[3] |
C. Bu, X. Jin, H. Li and C. Deng,
Brauer-type eigenvalue inclusion sets and the spectral radius of tensors, Linear Algebra Appl., 512 (2017), 234-248.
doi: 10.1016/j.laa.2016.09.041. |
[4] |
W. Ding and Y. Wei,
Solving multi-linear systems with M-tensors, J. Sci. Comput., 68 (2016), 689-715.
doi: 10.1007/s10915-015-0156-7. |
[5] |
F. Fang,
Bounds on eigenvalues of Hadamard product and the Fan product of matrices, Linear Algebra Appl., 425 (2007), 7-15.
doi: 10.1016/j.laa.2007.03.024. |
[6] |
S. Friedland, S. Gaubert and L. Han,
Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl., 438 (2013), 738-749.
doi: 10.1016/j.laa.2011.02.042. |
[7] |
L. Gao, D. Wang and G. Wang,
Further results on exponential stability for impulsive switched nonlinear time-delay systems with delayed impulse effects, Appl. Math. Comput., (2015), 186-200.
doi: 10.1016/j.amc.2015.06.023. |
[8] |
L. Gao and D. Wang,
Input-to-state stability and integral inputto-state stability for impulsive switched systems with time-delay under asynchronous switching, Nonlinear Anal.-Hybri., (2016), 55-71.
doi: 10.1016/j.nahs.2015.12.002. |
[9] |
R. Horn and C. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1985.
doi: 10.1017/CBO9780511810817. |
[10] |
C. Jutten and J. Herault,
Blind separation of sources, part Ⅰ: An adaptive algorithm based on neurmimetic architecture, Signal Process., 24 (1991), 1-10.
|
[11] |
L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, Mexico (2005), 129-132. |
[12] |
Y. Li, F. Chen and D. Wang,
New lower bounds on eigenvalue of the Hadamard product of an M-matrix and its inverse, Linear Algebra Appl., 430 (2009), 1423-1431.
doi: 10.1016/j.laa.2008.11.002. |
[13] |
Q. Liu, G. Chen and L. Zhao,
Some new bounds on the spectral radius of matrices, Linear Algebra Appl., 432 (2010), 936-948.
doi: 10.1016/j.laa.2009.10.006. |
[14] |
M. Ng, L. Qi and G. Zhou,
Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.
doi: 10.1137/09074838X. |
[15] |
Q. Ni, L. Qi and F. Wang,
An eigenvalue method for testing the positive definiteness of a multivariate form, IEEE Trans. Automat. Contr., 53 (2008), 1096-1107.
doi: 10.1109/TAC.2008.923679. |
[16] |
L. Qi,
Eigenvalues of an even-order real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
[17] |
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. |
[18] |
L. Sun, B. Zheng, J. Zhou and H. Yan,
Some inequalities for the Hadamard product of tensors, Linear Multilinear Algebra, 66 (2018), 1199-1214.
doi: 10.1080/03081087.2017.1346060. |
[19] |
G. Wang, G. Zhou and L. Caccetta,
Z-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst. Ser-B., 22 (2017), 187-198.
doi: 10.3934/dcdsb.2017009. |
[20] |
G. Wang, Y. Wang and Y. Wang,
Some Ostrowski-type bound estimations of spectral radius for weakly irreducible nonnegative tensors, Linear Multilinear Algebra, (2019).
doi: 10.1080/03081087.2018.1561823. |
[21] |
G. Wang, Y. Wang and L. Liu,
Bound estimations on the eigenvalues for Fan product of M-tensors, Taiwan. J. Math., 23 (2019), 751-766.
doi: 10.11650/tjm/180905. |
[22] |
G. Wang, Y. Wang and Y. Zhang,
Some inequalities for the Fan product of M-tensors, J. Inequal. Appl., 257 (2018), 15 pp.
doi: 10.1186/s13660-018-1853-1. |
[23] |
G. Wang, G. Zhou and L. Caccetta,
Sharp Brauer-type eigenvalue inclusion theorems for tensors, Pac. J. Optim., 14 (2018), 227-244.
|
[24] |
X. Wang, H. Chen and Y. Wang,
Solution structures of tensor complementarity problem, Front. Math. China., 13 (2018), 935-945.
doi: 10.1007/s11464-018-0675-2. |
[25] |
Y. Wang, G. Zhou and L. Caccetta,
Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Linear. Algebra. Appl., 22 (2015), 1059-1076.
doi: 10.1002/nla.1996. |
[26] |
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. |
[27] |
Y. Yang and Q. Yang,
Further results for Perron-Frobenius theorem for nonnegative tensors Ⅰ, SIAM J. Matrix Anal. Appl., 31 (2010), 2517-2530.
doi: 10.1137/090778766. |
[28] |
L. Zhang, L. Qi and G. Zhou,
M-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437-452.
doi: 10.1137/130915339. |
[29] |
D. Zhou, G. Chen, G. Wu and X. Zhang,
On some new bounds for eigenvalues of the Hadamard product and the Fan product of matrices, Linear Algebra Appl., 438 (2013), 1415-1426.
doi: 10.1016/j.laa.2012.09.013. |
[30] |
G. Zhou, G. Wang, L. Qi and A. Alqahtani,
A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer. Linear. Algebra. Appl., 25 (2018), e2134.
doi: 10.1002/nla.2134. |
[31] |
J. Zhou, L. Sun, L. P. Wei and C. Bu,
Some characterizations of M-tensors via digraphs, Linear Algebra Appl., 495 (2016), 190-198.
doi: 10.1016/j.laa.2016.01.041. |
show all references
References:
[1] |
L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, in Medical Image Computing and Computer-Assisted Intervention, Springer, 2008, 1-8.
doi: 10.1007/978-3-540-85988-8_1. |
[2] |
C. Bu, Y. Wei, L. Sun and J. Zhou,
Brualdi-type eigenvalue inclusion sets of tensors, Linear Algebra Appl., 480 (2015), 168-175.
doi: 10.1016/j.laa.2015.04.034. |
[3] |
C. Bu, X. Jin, H. Li and C. Deng,
Brauer-type eigenvalue inclusion sets and the spectral radius of tensors, Linear Algebra Appl., 512 (2017), 234-248.
doi: 10.1016/j.laa.2016.09.041. |
[4] |
W. Ding and Y. Wei,
Solving multi-linear systems with M-tensors, J. Sci. Comput., 68 (2016), 689-715.
doi: 10.1007/s10915-015-0156-7. |
[5] |
F. Fang,
Bounds on eigenvalues of Hadamard product and the Fan product of matrices, Linear Algebra Appl., 425 (2007), 7-15.
doi: 10.1016/j.laa.2007.03.024. |
[6] |
S. Friedland, S. Gaubert and L. Han,
Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl., 438 (2013), 738-749.
doi: 10.1016/j.laa.2011.02.042. |
[7] |
L. Gao, D. Wang and G. Wang,
Further results on exponential stability for impulsive switched nonlinear time-delay systems with delayed impulse effects, Appl. Math. Comput., (2015), 186-200.
doi: 10.1016/j.amc.2015.06.023. |
[8] |
L. Gao and D. Wang,
Input-to-state stability and integral inputto-state stability for impulsive switched systems with time-delay under asynchronous switching, Nonlinear Anal.-Hybri., (2016), 55-71.
doi: 10.1016/j.nahs.2015.12.002. |
[9] |
R. Horn and C. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1985.
doi: 10.1017/CBO9780511810817. |
[10] |
C. Jutten and J. Herault,
Blind separation of sources, part Ⅰ: An adaptive algorithm based on neurmimetic architecture, Signal Process., 24 (1991), 1-10.
|
[11] |
L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, Mexico (2005), 129-132. |
[12] |
Y. Li, F. Chen and D. Wang,
New lower bounds on eigenvalue of the Hadamard product of an M-matrix and its inverse, Linear Algebra Appl., 430 (2009), 1423-1431.
doi: 10.1016/j.laa.2008.11.002. |
[13] |
Q. Liu, G. Chen and L. Zhao,
Some new bounds on the spectral radius of matrices, Linear Algebra Appl., 432 (2010), 936-948.
doi: 10.1016/j.laa.2009.10.006. |
[14] |
M. Ng, L. Qi and G. Zhou,
Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.
doi: 10.1137/09074838X. |
[15] |
Q. Ni, L. Qi and F. Wang,
An eigenvalue method for testing the positive definiteness of a multivariate form, IEEE Trans. Automat. Contr., 53 (2008), 1096-1107.
doi: 10.1109/TAC.2008.923679. |
[16] |
L. Qi,
Eigenvalues of an even-order real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
[17] |
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. |
[18] |
L. Sun, B. Zheng, J. Zhou and H. Yan,
Some inequalities for the Hadamard product of tensors, Linear Multilinear Algebra, 66 (2018), 1199-1214.
doi: 10.1080/03081087.2017.1346060. |
[19] |
G. Wang, G. Zhou and L. Caccetta,
Z-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst. Ser-B., 22 (2017), 187-198.
doi: 10.3934/dcdsb.2017009. |
[20] |
G. Wang, Y. Wang and Y. Wang,
Some Ostrowski-type bound estimations of spectral radius for weakly irreducible nonnegative tensors, Linear Multilinear Algebra, (2019).
doi: 10.1080/03081087.2018.1561823. |
[21] |
G. Wang, Y. Wang and L. Liu,
Bound estimations on the eigenvalues for Fan product of M-tensors, Taiwan. J. Math., 23 (2019), 751-766.
doi: 10.11650/tjm/180905. |
[22] |
G. Wang, Y. Wang and Y. Zhang,
Some inequalities for the Fan product of M-tensors, J. Inequal. Appl., 257 (2018), 15 pp.
doi: 10.1186/s13660-018-1853-1. |
[23] |
G. Wang, G. Zhou and L. Caccetta,
Sharp Brauer-type eigenvalue inclusion theorems for tensors, Pac. J. Optim., 14 (2018), 227-244.
|
[24] |
X. Wang, H. Chen and Y. Wang,
Solution structures of tensor complementarity problem, Front. Math. China., 13 (2018), 935-945.
doi: 10.1007/s11464-018-0675-2. |
[25] |
Y. Wang, G. Zhou and L. Caccetta,
Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Linear. Algebra. Appl., 22 (2015), 1059-1076.
doi: 10.1002/nla.1996. |
[26] |
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. |
[27] |
Y. Yang and Q. Yang,
Further results for Perron-Frobenius theorem for nonnegative tensors Ⅰ, SIAM J. Matrix Anal. Appl., 31 (2010), 2517-2530.
doi: 10.1137/090778766. |
[28] |
L. Zhang, L. Qi and G. Zhou,
M-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437-452.
doi: 10.1137/130915339. |
[29] |
D. Zhou, G. Chen, G. Wu and X. Zhang,
On some new bounds for eigenvalues of the Hadamard product and the Fan product of matrices, Linear Algebra Appl., 438 (2013), 1415-1426.
doi: 10.1016/j.laa.2012.09.013. |
[30] |
G. Zhou, G. Wang, L. Qi and A. Alqahtani,
A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer. Linear. Algebra. Appl., 25 (2018), e2134.
doi: 10.1002/nla.2134. |
[31] |
J. Zhou, L. Sun, L. P. Wei and C. Bu,
Some characterizations of M-tensors via digraphs, Linear Algebra Appl., 495 (2016), 190-198.
doi: 10.1016/j.laa.2016.01.041. |
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