American Institute of Mathematical Sciences

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April  2019, 15(2): 507-516. doi: 10.3934/jimo.2018054

Exclusion sets in the Δ-type eigenvalue inclusion set for tensors

Yaotang Li , and  Suhua Li

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

* Corresponding author: Yaotang Li

Received  August 2017 Revised  October 2017 Published  April 2019 Early access  April 2018

Fund Project: The first author is supported by National Natural Science Foundations of China (11361074).

By excluding some sets which don't include any eigenvalue of a given tensor from the Δ-type eigenvalue inclusion set, two new Δ-type eigenvalue inclusion sets of tensors are given. And two criteria for identifying nonsingular tensors are also provided by using the new Δ-type eigenvalue inclusion sets.

Keywords: Tensors, eigenvalues, Δ-type eigenvalue inclusion set, exclusion sets.
Mathematics Subject Classification: Primary: 15A18, 15A69.
Citation: Yaotang Li, Suhua Li. Exclusion sets in the Δ-type eigenvalue inclusion set for tensors. Journal of Industrial & Management Optimization, 2019, 15 (2) : 507-516. doi: 10.3934/jimo.2018054
References:
[1]

C. Bu, Y. Wei, L. Sun and J. Zhou, Brualdi-type eigenvalue inclusion sets of tensors, Linear Algebra and its Applications, 480 (2015), 168-175.  doi: 10.1016/j.laa.2015.04.034.  Google Scholar

[2]

C. J. Hillar and L. -H. Lim, Most tensor problems are NP-hard, Journal of the ACM (JACM), 60 (2013), Art. 45, 39 pp.  Google Scholar

[3]

S. Hu, Z. Huang, C. Ling and L. Qi, On determinants and eigenvalue theory of tensors, Journal of Symbolic Computation, 50 (2013), 508-531.  doi: 10.1016/j.jsc.2012.10.001.  Google Scholar

[4]

Z. Huang, L. Wang, Z. Xu and J. Cui, A new S-type eigenvalue inclusion set for tensors and its applications, Journal of Inequalities and Applications, 2016 (2016), Paper No. 254, 19 pp.  Google Scholar

[5]

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

[6]

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

[7]

C. Li, Z. Chen and Y. Li, A new eigenvalue inclusion set for tensors and its applications, Linear Algebra and its Applications, 481 (2015), 36-53.  doi: 10.1016/j.laa.2015.04.023.  Google Scholar

[8]

C. Li, A. Jiao and Y. Li, An S-type eigenvalue localization set for tensors, Linear Algebra and its Applications, 493 (2016), 469-483.  doi: 10.1016/j.laa.2015.12.018.  Google Scholar

[9]

C. Li, J. Zhou and Y. Li, A new Brauer-type eigenvalue localization set for tensors, Linear and Multilinear Algebra, 64 (2016), 727-736.  doi: 10.1080/03081087.2015.1119779.  Google Scholar

[10]

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 MultiSensor Adaptive Processing, 2005,129–132. Google Scholar

[11]

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

[12]

L. Qi, Eigenvalues of a Supersymmetric Tensor and Positive Definiteness of an Even Degree Multivariate Form, Department of Applied Mathematics, The Hong Kong Polytechnic University, 2004. Google Scholar

[13]

C. Sang and J. Zhao, A new eigenvalue inclusion set for tensors with its applications, Cogent Mathematics, 4 (2017), 1320831. doi: 10.1080/23311835.2017.1320831.  Google Scholar

[14]

X. Wang and Y. Wei, H-tensors and nonsingular H-tensors, Frontiers of Mathematics in China, 11 (2016), 557-575.  doi: 10.1007/s11464-015-0495-6.  Google Scholar

[15]

Y. Yang and Q. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 2517-2530.  doi: 10.1137/090778766.  Google Scholar

[16]

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

show all references

References:
[1]

C. Bu, Y. Wei, L. Sun and J. Zhou, Brualdi-type eigenvalue inclusion sets of tensors, Linear Algebra and its Applications, 480 (2015), 168-175.  doi: 10.1016/j.laa.2015.04.034.  Google Scholar

[2]

C. J. Hillar and L. -H. Lim, Most tensor problems are NP-hard, Journal of the ACM (JACM), 60 (2013), Art. 45, 39 pp.  Google Scholar

[3]

S. Hu, Z. Huang, C. Ling and L. Qi, On determinants and eigenvalue theory of tensors, Journal of Symbolic Computation, 50 (2013), 508-531.  doi: 10.1016/j.jsc.2012.10.001.  Google Scholar

[4]

Z. Huang, L. Wang, Z. Xu and J. Cui, A new S-type eigenvalue inclusion set for tensors and its applications, Journal of Inequalities and Applications, 2016 (2016), Paper No. 254, 19 pp.  Google Scholar

[5]

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

[6]

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

[7]

C. Li, Z. Chen and Y. Li, A new eigenvalue inclusion set for tensors and its applications, Linear Algebra and its Applications, 481 (2015), 36-53.  doi: 10.1016/j.laa.2015.04.023.  Google Scholar

[8]

C. Li, A. Jiao and Y. Li, An S-type eigenvalue localization set for tensors, Linear Algebra and its Applications, 493 (2016), 469-483.  doi: 10.1016/j.laa.2015.12.018.  Google Scholar

[9]

C. Li, J. Zhou and Y. Li, A new Brauer-type eigenvalue localization set for tensors, Linear and Multilinear Algebra, 64 (2016), 727-736.  doi: 10.1080/03081087.2015.1119779.  Google Scholar

[10]

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 MultiSensor Adaptive Processing, 2005,129–132. Google Scholar

[11]

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

[12]

L. Qi, Eigenvalues of a Supersymmetric Tensor and Positive Definiteness of an Even Degree Multivariate Form, Department of Applied Mathematics, The Hong Kong Polytechnic University, 2004. Google Scholar

[13]

C. Sang and J. Zhao, A new eigenvalue inclusion set for tensors with its applications, Cogent Mathematics, 4 (2017), 1320831. doi: 10.1080/23311835.2017.1320831.  Google Scholar

[14]

X. Wang and Y. Wei, H-tensors and nonsingular H-tensors, Frontiers of Mathematics in China, 11 (2016), 557-575.  doi: 10.1007/s11464-015-0495-6.  Google Scholar

[15]

Y. Yang and Q. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 2517-2530.  doi: 10.1137/090778766.  Google Scholar

[16]

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

Figure 1.  $C(\mathcal{A}_{0})\nsubseteqq V(\mathcal{A}_{0})$ and $C(\mathcal{A}_{0})\nsupseteqq V(\mathcal{A}_{0})$.
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Figure 2.  $C(\mathcal{A}_{1})\subset \Theta(\mathcal{A}_{1})$.
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Figure 3.  $V(\mathcal{A}_{2})\subset \Theta(\mathcal{A}_{2})$.
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