doi: 10.3934/jimo.2019042

$ E $-eigenvalue localization sets for tensors

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  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.

Citation: Caili Sang, Zhen Chen. $ E $-eigenvalue localization sets for tensors. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019042
References:
[1]

A. AmmarF. 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.

[2]

B. D. AndersonN. 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.

[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.

[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.

[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.

[6]

K. C. ChangK. 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.

[7]

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

[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.

[9]

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

[10]

J. HeY. M. LiuH. KeJ. 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.

[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.

[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.

[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.

[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.

[15]

L. D. LathauwerB. 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.

[16]

W. LiD. 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.

[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.

[18]

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.

[19]

C. LiZ. 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.

[20]

C. LiJ. 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.

[21]

C. LiA. 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.

[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.

[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.

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[31]

G. WangG. 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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

A. AmmarF. 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.

[2]

B. D. AndersonN. 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.

[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.

[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.

[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.

[6]

K. C. ChangK. 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.

[7]

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

[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.

[9]

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

[10]

J. HeY. M. LiuH. KeJ. 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.

[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.

[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.

[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.

[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.

[15]

L. D. LathauwerB. 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.

[16]

W. LiD. 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.

[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.

[18]

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.

[19]

C. LiZ. 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.

[20]

C. LiJ. 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.

[21]

C. LiA. 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.

[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.

[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.

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[31]

G. WangG. 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.

[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.

[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.

[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.

[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.

Figure 1.  Comparisons of $ \mathcal{K}(\mathcal{A}) $, $ \mathcal{L}(\mathcal{A}) $, $ \Psi(\mathcal{A}) $, $ \Upsilon(\mathcal{A}) $ and $ \Omega(\mathcal{A}) $.
Figure 2.  Comparisons of $ \Omega(\mathcal{A}) $ and $ \triangle(\mathcal{A}) $.
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
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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