# American Institute of Mathematical Sciences

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

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

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

* Corresponding author: Yaotang Li

Received  August 2017 Revised  October 2017 Published  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.

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