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June  2018, 8(2): 203-210. doi: 10.3934/naco.2018012

New bounds for eigenvalues of strictly diagonally dominant tensors

Yining Gu and  Wei Wu

School of Mathematical Sciences, Tianjin University, Tianjin 300350, China

* Corresponding author: Wei Wu

Received  February 2017 Revised  March 2018 Published  May 2018

Fund Project: This work was supported by NSF grant of China (Grant No. 11371276).

In this paper, we prove that the minimum eigenvalue of a strictly diagonally dominant Z-tensor with positive diagonal entries lies between the smallest and the largest row sums. The novelty comes from the upper bound. Moreover, we show that a similar upper bound does not hold for the minimum eigenvalue of a strictly diagonally dominant tensor with positive diagonal entries but with arbitrary off-diagonal entries. Furthermore, other new bounds for the minimum eigenvalue of nonsingular M-tensors are obtained.

Keywords: Diagonal dominance, M-tensor, minimum eigenvalue.
Mathematics Subject Classification: Primary: 15A18, 15A69, 65F15.
Citation: Yining Gu, Wei Wu. New bounds for eigenvalues of strictly diagonally dominant tensors. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 203-210. doi: 10.3934/naco.2018012
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-MICCAI 2008, Springer, Berlin/Heidelberg, (2008), 1-8. Google Scholar

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

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K. C. Chang, K. Pearson and T. Zhang, Perron Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.   Google Scholar

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W Ding, L. Qi and Y. Wei, M-Tensors and nonsingular M-Tensors, Linear Algebra and Its Applications, 439 (2013), 3264-3278.  doi: 10.1016/j.laa.2013.08.038.  Google Scholar

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J. He and Z. Huang, Upper bound for the largest Z-eigenvalue of positive tensors, Linear Algebra and Its Applications, 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.  Google Scholar

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J. He and Z. Huang, Inequalities for M-tensors Journal of Inequalities and Applications, (2014), 114, 9 pages. doi: 10.1186/1029-242X-2014-114.  Google Scholar

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L. De Lathauwer, B. D. Moor and J. Vandewalle, On the best rank-1 and rank-($R_{1},R_{2},...,R_{N}$) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.  doi: 10.1137/S0895479898346995.  Google Scholar

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C. Li, Y. Li and K. Xu, New eigenvalue inclusion sets for tensors, Numerical Linear Algebra with Applications, 21 (2014), 39-50.  doi: 10.1002/nla.1858.  Google Scholar

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L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in CAMSAP'05: Proceeding of the IEEE International Workshop on Computational Advances in Multi-tensor Adaptive Processing, (2005), 129-132.   Google Scholar

[10]

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

[11]

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

[12]

L. Qi, W. Sun and Y. Wang, Numerical multilinear algebra and its applications, Front. Math. China, 2 (2007), 501-526.  doi: 10.1007/s11464-007-0031-4.  Google Scholar

[13]

L. Qi, Y. Wang and E. X. Wu, D-eigenvalues of diffusion kurtosis tensor, Journal of Computational and Applied Mathematics, 221 (2008), 150-157.  doi: 10.1016/j.cam.2007.10.012.  Google Scholar

[14]

F. Wang, The tensor eigenvalue methods for the positive definiteness identification problem, Available at http://ira.lib.polyu.edu.hk/handle/10397/2642, Hong Kong Polytechnic University, 2006.  Google Scholar

[15]

Q. Yang and Y. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors Ⅱ, SIAM Journal on Matrix Analysis and Applications, 32 (2011), 1236-1250.  doi: 10.1137/100813671.  Google Scholar

[16]

Y. Yang and Q. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM Journal on Matrix Analysis and Applications, 31 (2011), 2517-2530.  doi: 10.1137/090778766.  Google Scholar

[17]

L. Zhang, L. Qi and G. Zhou, M-tensors and the positive definiteness of a multivariate form, Mathematics, 2012. Google Scholar

[18]

L. Zhang, L. Qi and G. Zhou, M-tensors and some applications, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 437-452.  doi: 10.1137/130915339.  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-MICCAI 2008, Springer, Berlin/Heidelberg, (2008), 1-8. Google Scholar

[2]

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

[3]

K. C. Chang, K. Pearson and T. Zhang, Perron Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.   Google Scholar

[4]

W Ding, L. Qi and Y. Wei, M-Tensors and nonsingular M-Tensors, Linear Algebra and Its Applications, 439 (2013), 3264-3278.  doi: 10.1016/j.laa.2013.08.038.  Google Scholar

[5]

J. He and Z. Huang, Upper bound for the largest Z-eigenvalue of positive tensors, Linear Algebra and Its Applications, 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.  Google Scholar

[6]

J. He and Z. Huang, Inequalities for M-tensors Journal of Inequalities and Applications, (2014), 114, 9 pages. doi: 10.1186/1029-242X-2014-114.  Google Scholar

[7]

L. De Lathauwer, B. D. Moor and J. Vandewalle, On the best rank-1 and rank-($R_{1},R_{2},...,R_{N}$) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.  doi: 10.1137/S0895479898346995.  Google Scholar

[8]

C. Li, Y. Li and K. Xu, New eigenvalue inclusion sets for tensors, Numerical Linear Algebra with Applications, 21 (2014), 39-50.  doi: 10.1002/nla.1858.  Google Scholar

[9]

L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in CAMSAP'05: Proceeding of the IEEE International Workshop on Computational Advances in Multi-tensor Adaptive Processing, (2005), 129-132.   Google Scholar

[10]

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

[11]

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

[12]

L. Qi, W. Sun and Y. Wang, Numerical multilinear algebra and its applications, Front. Math. China, 2 (2007), 501-526.  doi: 10.1007/s11464-007-0031-4.  Google Scholar

[13]

L. Qi, Y. Wang and E. X. Wu, D-eigenvalues of diffusion kurtosis tensor, Journal of Computational and Applied Mathematics, 221 (2008), 150-157.  doi: 10.1016/j.cam.2007.10.012.  Google Scholar

[14]

F. Wang, The tensor eigenvalue methods for the positive definiteness identification problem, Available at http://ira.lib.polyu.edu.hk/handle/10397/2642, Hong Kong Polytechnic University, 2006.  Google Scholar

[15]

Q. Yang and Y. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors Ⅱ, SIAM Journal on Matrix Analysis and Applications, 32 (2011), 1236-1250.  doi: 10.1137/100813671.  Google Scholar

[16]

Y. Yang and Q. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM Journal on Matrix Analysis and Applications, 31 (2011), 2517-2530.  doi: 10.1137/090778766.  Google Scholar

[17]

L. Zhang, L. Qi and G. Zhou, M-tensors and the positive definiteness of a multivariate form, Mathematics, 2012. Google Scholar

[18]

L. Zhang, L. Qi and G. Zhou, M-tensors and some applications, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 437-452.  doi: 10.1137/130915339.  Google Scholar

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