American Institute of Mathematical Sciences

Advanced
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  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 & Management Optimization, 2020, 16 (5) : 2551-2562. doi: 10.3934/jimo.2019069
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[23]

G. Wang, G. Zhou and L. Caccetta, Sharp Brauer-type eigenvalue inclusion theorems for tensors, Pac. J. Optim., 14 (2018), 227-244.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

G. Wang, G. Zhou and L. Caccetta, Sharp Brauer-type eigenvalue inclusion theorems for tensors, Pac. J. Optim., 14 (2018), 227-244.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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