American Institute of Mathematical Sciences

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September  2020, 16(5): 2551-2562. doi: 10.3934/jimo.2019069

Brualdi-type inequalities on the minimum eigenvalue for the Fan product of M-tensors

Gang Wang , , Yiju Wang and  Yuan Zhang

School of Management Science, Qufu Normal University, Rizhao, Shandong 276826, China

* Corresponding author: Gang Wang

Received  September 2018 Revised  March 2019 Published  September 2020 Early access  July 2019

Fund Project: This work was supported by the Natural Science Foundation of China (11671228) and the Natural Science Foundation of Shandong Province (ZR2016AM10)

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.

Keywords: Fan product, nonnegative tensors, $ M $-tensors.
Mathematics Subject Classification: Primary: 15A18, 15A42.
Citation: Gang Wang, Yiju Wang, Yuan Zhang. Brualdi-type inequalities on the minimum eigenvalue for the Fan product of M-tensors. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2551-2562. doi: 10.3934/jimo.2019069
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.

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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.

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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.

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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.

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R. Horn and C. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1985. doi: 10.1017/CBO9780511810817.

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C. Jutten and J. Herault, Blind separation of sources, part Ⅰ: An adaptive algorithm based on neurmimetic architecture, Signal Process., 24 (1991), 1-10. 

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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.

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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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