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Exclusion sets in the Δ-type eigenvalue inclusion set for tensors
School of Mathematics and Statistics, Yunnan University, Kunming 650091, China |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |



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