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July  2020, 16(4): 2045-2063. doi: 10.3934/jimo.2019042

$ E $-eigenvalue localization sets for tensors

Caili Sang 1,2, and  Zhen Chen 1,,

1. 

School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou 550025, China
2. 

College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025, China

* Corresponding author: Zhen Chen

Received  September 2018 Revised  December 2018 Published  July 2020 Early access  May 2019

Fund Project: This work is supported by National Natural Science Foundation of China (No. 11501141), Science and Technology Projects of Education Department of Guizhou Province (Grant No. KY[2015]352), and Science and Technology Top-notch Talents Support Project of Education Department of Guizhou Province (Grant No. QJHKYZ [2016]066)

Several existing $Z$-eigenvalue localization sets for tensors are first generalized to $E$-eigenvalue localization sets. And then two tighter $ E$-eigenvalue localization sets for tensors are presented. As applications, a sufficient condition for the positive definiteness of fourth-order real symmetric tensors, a sufficient condition for the positive semi-definiteness of fourth-order real symmetric tensors, and a new upper bound for the $ Z$-spectral radius of weakly symmetric nonnegative tensors are obtained. Finally, numerical examples are given to verify the theoretical results.

Keywords: Z-eigenvalues, E-eigenvalues, localization sets, nonnegative tensors, spectral radius.
Mathematics Subject Classification: Primary: 15A18, 15A69, 15A72, 15A42.
Citation: Caili Sang, Zhen Chen. $ E $-eigenvalue localization sets for tensors. Journal of Industrial & Management Optimization, 2020, 16 (4) : 2045-2063. doi: 10.3934/jimo.2019042
References:
[1]

A. Ammar, F. Chinesta and A. Falcó, On the convergence of a greedy rank-one update algorithm for a class of linear systems, Arch. Comput. Methods Eng., 17 (2010), 473-486.  doi: 10.1007/s11831-010-9048-z.  Google Scholar

[2]

B. D. Anderson, N. K. Bose and E. I. Jury, Output feedback stabilization and related problems-solutions via decision methods, IEEE Trans. Automat. Control, AC20 (1975), 53-66.  doi: 10.1109/tac.1975.1100846.  Google Scholar

[3]

L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, Med. Image Comput. Comput. Assist. Interv., 5241 (2008), 1–8. Available from: https://www.ncbi.nlm.nih.gov/pubmed/18979725. doi: 10.1007/978-3-540-85988-8_1.  Google Scholar

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N. K. Bose and P. S. Kamt, Algorithm for stability test of multidimensional filters, IEEE Trans. Acoust. Speech Signal Process, 22 (1974), 307-314.  doi: 10.1109/TASSP.1974.1162592.  Google Scholar

[5]

N. K. Bose and R. W. Newcomb, Tellegon's theorem and multivariate realizability theory, Int. J. Electron, 36 (1974), 417-425.  doi: 10.1080/00207217408900421.  Google Scholar

[6]

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

[7]

R. A. Devore and V. N. Temlyakov, Some remarks on greedy algorithms, Adv. comput. Math., 5 (1996), 173-187.  doi: 10.1007/BF02124742.  Google Scholar

[8]

A. Falco and A. Nouy, A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach, J. Math. Anal. Appl., 376 (2011), 469-480.  doi: 10.1016/j.jmaa.2010.12.003.  Google Scholar

[9]

J. He, Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl., 20 (2016), 1290-1301.   Google Scholar

[10]

J. He, Y. M. Liu, H. Ke, J. K. Tian and X. Li, Bounds for the $Z$-spectral radius of nonnegative tensors, Springerplus, 5 (2016), 1727.  doi: 10.1186/s40064-016-3338-3.  Google Scholar

[11]

J. He and T. Z. Huang, Upper bound for the largest $Z$-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.  Google Scholar

[12]

J. C. Hsu and A. U. Meyer, Modern Control Principles and Applications, The McGraw-Hill Series in Advanced Chemistry McGraw-Hill Book Co., Inc., New York-Toronto-London, 1956.  Google Scholar

[13]

E. Kofidis and P. A. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884.  doi: 10.1137/S0895479801387413.  Google Scholar

[14]

T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), 1095-1124.  doi: 10.1137/100801482.  Google Scholar

[15]

L. D. Lathauwer, B. D. Moor and J. Vandewalle, On the best rank-1 and rank-($R_1,R_2,\ldots ,R_N $) approximation of higer-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.  doi: 10.1137/S0895479898346995.  Google Scholar

[16]

W. Li, D. Liu and S. W. Vong, $Z$-eigenpair bounds for an irreducible nonnegative tensor, Linear Algebra Appl., 483 (2015), 182-199.  doi: 10.1016/j.laa.2015.05.033.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

Q. Liu and Y. Li, Bounds for the $ Z$-eigenpair of general nonnegative tensors, Open Math., 14 (2016), 181-194.  doi: 10.1515/math-2016-0017.  Google Scholar

[24]

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

[25]

L. Qi, G. Yu and E. X. Wu, Higher order positive semidefinite diffusion tensor imaging, SIAM J. Imaging Sciences, 3 (2010), 416-433.  doi: 10.1137/090755138.  Google Scholar

[26]

L. Qi, Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines, J. Symbolic Comput., 41 (2006), 1309-1327.  doi: 10.1016/j.jsc.2006.02.011.  Google Scholar

[27]

L. Qi, The best rank-one approximation ratio of a tensor space, SIAM J. Matrix Anal. Appl., 32 (2011), 430-442.  doi: 10.1137/100795802.  Google Scholar

[28]

C. Sang, A new Brauer-type $ Z$-eigenvalue inclusion set for tensors, Numer. Algor., (2018), 1–14. doi: 10.1007/s11075-018-0506-2.  Google Scholar

[29]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581-1595.  doi: 10.1137/130909135.  Google Scholar

[30]

Y. Wang and G. Wang, Two $ S$-type $ Z$-eigenvalue inclusion sets for tensors, J. Inequal. Appl., 2017 (2017), Paper No. 152, 12 pp. doi: 10.1186/s13660-017-1428-6.  Google Scholar

[31]

G. Wang, G. L. Zhou and L. Caccetta, $Z$-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst., Ser. B., 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.  Google Scholar

[32]

Y. Wang and L. Qi, On the successive supersymmetric rank-1 decomposition of higher-order supersymmetric tensors, Numer. Linear Algebra Appl., 14 (2007), 503-519.  doi: 10.1002/nla.537.  Google Scholar

[33]

T. Zhang and G. H. Golub, Rank-one approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.  doi: 10.1137/S0895479899352045.  Google Scholar

[34]

J. Zhao, A new $ Z$-eigenvalue localization set for tensors, J. Inequal. Appl., 2017 (2017), Paper No. 85, 9 pp. doi: 10.1186/s13660-017-1363-6.  Google Scholar

[35]

J. Zhao and C. Sang, Two new eigenvalue localization sets for tensors and theirs applications, Open Math., 15 (2017), 1267-1276.  doi: 10.1515/math-2017-0106.  Google Scholar

show all references

References:
[1]

A. Ammar, F. Chinesta and A. Falcó, On the convergence of a greedy rank-one update algorithm for a class of linear systems, Arch. Comput. Methods Eng., 17 (2010), 473-486.  doi: 10.1007/s11831-010-9048-z.  Google Scholar

[2]

B. D. Anderson, N. K. Bose and E. I. Jury, Output feedback stabilization and related problems-solutions via decision methods, IEEE Trans. Automat. Control, AC20 (1975), 53-66.  doi: 10.1109/tac.1975.1100846.  Google Scholar

[3]

L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, Med. Image Comput. Comput. Assist. Interv., 5241 (2008), 1–8. Available from: https://www.ncbi.nlm.nih.gov/pubmed/18979725. doi: 10.1007/978-3-540-85988-8_1.  Google Scholar

[4]

N. K. Bose and P. S. Kamt, Algorithm for stability test of multidimensional filters, IEEE Trans. Acoust. Speech Signal Process, 22 (1974), 307-314.  doi: 10.1109/TASSP.1974.1162592.  Google Scholar

[5]

N. K. Bose and R. W. Newcomb, Tellegon's theorem and multivariate realizability theory, Int. J. Electron, 36 (1974), 417-425.  doi: 10.1080/00207217408900421.  Google Scholar

[6]

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

[7]

R. A. Devore and V. N. Temlyakov, Some remarks on greedy algorithms, Adv. comput. Math., 5 (1996), 173-187.  doi: 10.1007/BF02124742.  Google Scholar

[8]

A. Falco and A. Nouy, A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach, J. Math. Anal. Appl., 376 (2011), 469-480.  doi: 10.1016/j.jmaa.2010.12.003.  Google Scholar

[9]

J. He, Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl., 20 (2016), 1290-1301.   Google Scholar

[10]

J. He, Y. M. Liu, H. Ke, J. K. Tian and X. Li, Bounds for the $Z$-spectral radius of nonnegative tensors, Springerplus, 5 (2016), 1727.  doi: 10.1186/s40064-016-3338-3.  Google Scholar

[11]

J. He and T. Z. Huang, Upper bound for the largest $Z$-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.  Google Scholar

[12]

J. C. Hsu and A. U. Meyer, Modern Control Principles and Applications, The McGraw-Hill Series in Advanced Chemistry McGraw-Hill Book Co., Inc., New York-Toronto-London, 1956.  Google Scholar

[13]

E. Kofidis and P. A. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884.  doi: 10.1137/S0895479801387413.  Google Scholar

[14]

T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), 1095-1124.  doi: 10.1137/100801482.  Google Scholar

[15]

L. D. Lathauwer, B. D. Moor and J. Vandewalle, On the best rank-1 and rank-($R_1,R_2,\ldots ,R_N $) approximation of higer-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.  doi: 10.1137/S0895479898346995.  Google Scholar

[16]

W. Li, D. Liu and S. W. Vong, $Z$-eigenpair bounds for an irreducible nonnegative tensor, Linear Algebra Appl., 483 (2015), 182-199.  doi: 10.1016/j.laa.2015.05.033.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

Q. Liu and Y. Li, Bounds for the $ Z$-eigenpair of general nonnegative tensors, Open Math., 14 (2016), 181-194.  doi: 10.1515/math-2016-0017.  Google Scholar

[24]

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

[25]

L. Qi, G. Yu and E. X. Wu, Higher order positive semidefinite diffusion tensor imaging, SIAM J. Imaging Sciences, 3 (2010), 416-433.  doi: 10.1137/090755138.  Google Scholar

[26]

L. Qi, Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines, J. Symbolic Comput., 41 (2006), 1309-1327.  doi: 10.1016/j.jsc.2006.02.011.  Google Scholar

[27]

L. Qi, The best rank-one approximation ratio of a tensor space, SIAM J. Matrix Anal. Appl., 32 (2011), 430-442.  doi: 10.1137/100795802.  Google Scholar

[28]

C. Sang, A new Brauer-type $ Z$-eigenvalue inclusion set for tensors, Numer. Algor., (2018), 1–14. doi: 10.1007/s11075-018-0506-2.  Google Scholar

[29]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581-1595.  doi: 10.1137/130909135.  Google Scholar

[30]

Y. Wang and G. Wang, Two $ S$-type $ Z$-eigenvalue inclusion sets for tensors, J. Inequal. Appl., 2017 (2017), Paper No. 152, 12 pp. doi: 10.1186/s13660-017-1428-6.  Google Scholar

[31]

G. Wang, G. L. Zhou and L. Caccetta, $Z$-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst., Ser. B., 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.  Google Scholar

[32]

Y. Wang and L. Qi, On the successive supersymmetric rank-1 decomposition of higher-order supersymmetric tensors, Numer. Linear Algebra Appl., 14 (2007), 503-519.  doi: 10.1002/nla.537.  Google Scholar

[33]

T. Zhang and G. H. Golub, Rank-one approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.  doi: 10.1137/S0895479899352045.  Google Scholar

[34]

J. Zhao, A new $ Z$-eigenvalue localization set for tensors, J. Inequal. Appl., 2017 (2017), Paper No. 85, 9 pp. doi: 10.1186/s13660-017-1363-6.  Google Scholar

[35]

J. Zhao and C. Sang, Two new eigenvalue localization sets for tensors and theirs applications, Open Math., 15 (2017), 1267-1276.  doi: 10.1515/math-2017-0106.  Google Scholar

Figure 1.  Comparisons of $ \mathcal{K}(\mathcal{A}) $, $ \mathcal{L}(\mathcal{A}) $, $ \Psi(\mathcal{A}) $, $ \Upsilon(\mathcal{A}) $ and $ \Omega(\mathcal{A}) $.
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Figure 2.  Comparisons of $ \Omega(\mathcal{A}) $ and $ \triangle(\mathcal{A}) $.
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Table 1.  Upper bounds of $ \varrho(\mathcal{A}) $
Method $ \varrho(\mathcal{A})\leq $
Theorem 3.1, i.e., Corollary 4.5 of [29] 23.0000
Theorem 3.3 of [16] 22.8625
Theorem 3.4 of [35], where $ S=\{3\},\bar{S}=\{1,2\} $ 22.8521
Theorem 3.2, i.e., Theorem 4.5 of [31] 22.8149
Theorem 4.7 of [31] 22.7759
Theorem 2.9 of [23] 22.7217
Theorem 3.5 of [9] 22.7163
Theorem 4.6 of [31] 22.6478
Theorem 6 of [10] 22.6290
Theorem 3.3, i.e., Theorem 5 of [34] 22.5000
Theorem 4 of [30], where $ S=\{3\},\bar{S}=\{1,2\} $ 22.4195
Theorem 3.4, i.e., Theorem 7 of [28] 22.2122
Theorem 3.5 21.2604
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Method $ \varrho(\mathcal{A})\leq $
Theorem 3.1, i.e., Corollary 4.5 of [29] 23.0000
Theorem 3.3 of [16] 22.8625
Theorem 3.4 of [35], where $ S=\{3\},\bar{S}=\{1,2\} $ 22.8521
Theorem 3.2, i.e., Theorem 4.5 of [31] 22.8149
Theorem 4.7 of [31] 22.7759
Theorem 2.9 of [23] 22.7217
Theorem 3.5 of [9] 22.7163
Theorem 4.6 of [31] 22.6478
Theorem 6 of [10] 22.6290
Theorem 3.3, i.e., Theorem 5 of [34] 22.5000
Theorem 4 of [30], where $ S=\{3\},\bar{S}=\{1,2\} $ 22.4195
Theorem 3.4, i.e., Theorem 7 of [28] 22.2122
Theorem 3.5 21.2604
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