doi: 10.3934/jimo.2022106
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Generalized minimal Gershgorin set for tensors

1. 

Department of Mathematics, University of Hormozgan, P. O. Box 3995, Bandar Abbas, Iran

2. 

School of Mathematics and Statistics, Yunnan University, Kunming 650091, China

*Corresponding author: Mostafa Zangiabadi

Received  January 2022 Revised  March 2022 Early access June 2022

Generalized minimal Gershgorin set is introduced to locate all generalized tensor eigenvalues. It is proved that this set is tighter than the well-known generalized Gershgorin set. A sequence of approximation sets were also conducted in order to obtain the generalized minimal Gershgorin set and prove that the limit of this sequence is the generalized minimal Gershgorin set. Furthermore, we use numerical examples and figures to illustrate the benefits of our results.

Citation: Mohsen Tourang, Mostafa Zangiabadi, Chaoqian Li. Generalized minimal Gershgorin set for tensors. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022106
References:
[1]

K. C. ChangK. 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.

[2]

M. CheA. Cichocki and Y. Wei, Neural networks for computing best rank-one approximations of tensors and its applications, Neurocomputing, 267 (2017), 114-133. 

[3]

C. F. CuiY. H. Dai and J. Nie, All real eigenvalues of symmetric tensors, SIAM J. Matrix Anal. Appl., 35 (2014), 1582-1601.  doi: 10.1137/140962292.

[4]

W. DingZ. Hou and Y. Wei, Tensor logarithmic norm and its applications, Numer. Linear Algebra Appl., 23 (2016), 989-1006.  doi: 10.1002/nla.2064.

[5]

W. Ding and Y. Wei, Generalized tensor eigenvalue problems, SIAM J. Matrix Anal. Appl., 36 (2015), 1073-1099.  doi: 10.1137/140975656.

[6]

S. HuZ. H. HuangC. Ling and L. Qi, On determinants and eigenvalue theory of tensors, J. Symbolic Comput., 50 (2013), 508-531.  doi: 10.1016/j.jsc.2012.10.001.

[7]

T. G. Kolda and J. R. Mayo, An adaptive shifted power method for computing generalized tensor eigenpairs, SIAM J. Matrix Anal. Appl., 35 (2014), 1563-1581.  doi: 10.1137/140951758.

[8]

V. KostićL. J. Cvetković and R. S. Varga, Gershgorin-type localizations of generalized eigenvalues, Numer. Linear Algebra Appl., 16 (2009), 883-898.  doi: 10.1002/nla.671.

[9]

C. Li and Y. Li, An eigenvalue localization set for tensors with applications to determine the positive (semi-) definiteness of tensors, Linear Multilinear Algebra, 64 (2016), 587-601.  doi: 10.1080/03081087.2015.1049582.

[10]

C. LiC. Zhang and Y. Li, Minimal Gersgorin tensor eigenvalue inclusion set and its numerical approximation, J. Comput. Appl. Math., 302 (2016), 200-210.  doi: 10.1016/j.cam.2016.02.008.

[11]

C. LiY. Li and X. Kong, New eigenvalue inclusion sets for tensors, Numer. Linear Algebra Appl., 21 (2014), 39-50.  doi: 10.1002/nla.1858.

[12]

C. LiQ. Liu and Y. Wei, Pseudospectra localizations for generalized tensor eigenvalues to seek more positive definite tensors, Comput. Appl. Math., 38 (2019), 1-22.  doi: 10.1007/s40314-019-0958-6.

[13]

Y. LiuG. Zhou and N. F. Ibrahim, An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor, J. Comput. Appl. Math., 235 (2010), 286-292.  doi: 10.1016/j.cam.2010.06.002.

[14]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, 2017. doi: 10.1137/1.9781611974751. ch1.

[15]

L. Sun, S. Ji and J. Ye, Hypergraph spectral learning for multi-label classification, Proceedings of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Las Vegas, Nevada, USA, (2008), 668–676.

[16]

R. S. Varga, Gershgorin and his Circles, Springer Science and Business Media, 2010. doi: 10.1007/978-3-642-17798-9.

[17]

X. Wang and Y. Wei, H-tensors and nonsingular H-tensors, Front. Math. China, 11 (2016), 557-575.  doi: 10.1007/s11464-015-0495-6.

[18] Y. Wei and W. Ding, Theory and Computation of Tensors: Multi-Dimensional Arrays, Academic Press, 2016. 
[19]

Y. Yang and Q. Yang, Further results for Perron–Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl., 31 (2010), 2517-2530.  doi: 10.1137/090778766.

[20]

L. P. ZhangL. QiZ. Y. Luo and Y. Xu, The dominant eigenvalue of an essentially nonnegative tensor, Numer. Linear Algebra Appl., 20 (2013), 929-941.  doi: 10.1002/nla.1880.

show all references

References:
[1]

K. C. ChangK. 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.

[2]

M. CheA. Cichocki and Y. Wei, Neural networks for computing best rank-one approximations of tensors and its applications, Neurocomputing, 267 (2017), 114-133. 

[3]

C. F. CuiY. H. Dai and J. Nie, All real eigenvalues of symmetric tensors, SIAM J. Matrix Anal. Appl., 35 (2014), 1582-1601.  doi: 10.1137/140962292.

[4]

W. DingZ. Hou and Y. Wei, Tensor logarithmic norm and its applications, Numer. Linear Algebra Appl., 23 (2016), 989-1006.  doi: 10.1002/nla.2064.

[5]

W. Ding and Y. Wei, Generalized tensor eigenvalue problems, SIAM J. Matrix Anal. Appl., 36 (2015), 1073-1099.  doi: 10.1137/140975656.

[6]

S. HuZ. H. HuangC. Ling and L. Qi, On determinants and eigenvalue theory of tensors, J. Symbolic Comput., 50 (2013), 508-531.  doi: 10.1016/j.jsc.2012.10.001.

[7]

T. G. Kolda and J. R. Mayo, An adaptive shifted power method for computing generalized tensor eigenpairs, SIAM J. Matrix Anal. Appl., 35 (2014), 1563-1581.  doi: 10.1137/140951758.

[8]

V. KostićL. J. Cvetković and R. S. Varga, Gershgorin-type localizations of generalized eigenvalues, Numer. Linear Algebra Appl., 16 (2009), 883-898.  doi: 10.1002/nla.671.

[9]

C. Li and Y. Li, An eigenvalue localization set for tensors with applications to determine the positive (semi-) definiteness of tensors, Linear Multilinear Algebra, 64 (2016), 587-601.  doi: 10.1080/03081087.2015.1049582.

[10]

C. LiC. Zhang and Y. Li, Minimal Gersgorin tensor eigenvalue inclusion set and its numerical approximation, J. Comput. Appl. Math., 302 (2016), 200-210.  doi: 10.1016/j.cam.2016.02.008.

[11]

C. LiY. Li and X. Kong, New eigenvalue inclusion sets for tensors, Numer. Linear Algebra Appl., 21 (2014), 39-50.  doi: 10.1002/nla.1858.

[12]

C. LiQ. Liu and Y. Wei, Pseudospectra localizations for generalized tensor eigenvalues to seek more positive definite tensors, Comput. Appl. Math., 38 (2019), 1-22.  doi: 10.1007/s40314-019-0958-6.

[13]

Y. LiuG. Zhou and N. F. Ibrahim, An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor, J. Comput. Appl. Math., 235 (2010), 286-292.  doi: 10.1016/j.cam.2010.06.002.

[14]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, 2017. doi: 10.1137/1.9781611974751. ch1.

[15]

L. Sun, S. Ji and J. Ye, Hypergraph spectral learning for multi-label classification, Proceedings of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Las Vegas, Nevada, USA, (2008), 668–676.

[16]

R. S. Varga, Gershgorin and his Circles, Springer Science and Business Media, 2010. doi: 10.1007/978-3-642-17798-9.

[17]

X. Wang and Y. Wei, H-tensors and nonsingular H-tensors, Front. Math. China, 11 (2016), 557-575.  doi: 10.1007/s11464-015-0495-6.

[18] Y. Wei and W. Ding, Theory and Computation of Tensors: Multi-Dimensional Arrays, Academic Press, 2016. 
[19]

Y. Yang and Q. Yang, Further results for Perron–Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl., 31 (2010), 2517-2530.  doi: 10.1137/090778766.

[20]

L. P. ZhangL. QiZ. Y. Luo and Y. Xu, The dominant eigenvalue of an essentially nonnegative tensor, Numer. Linear Algebra Appl., 20 (2013), 929-941.  doi: 10.1002/nla.1880.

Figure 1.  $ \sigma (\mathcal{A},\mathcal{B}) \subseteq \Gamma^{\omega _4 } \left( {\mathcal{A},\mathcal{B}} \right) \subseteq \Gamma \left( {\mathcal{A},\mathcal{B}} \right) $
Figure 2.  $ \sigma (\mathcal{A},\mathcal{B}) \subseteq \Gamma^{\omega _4 } \left( {\mathcal{A},\mathcal{B}} \right) \subseteq \Gamma \left( {\mathcal{A},\mathcal{B}} \right) $
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