American Institute of Mathematical Sciences

Advanced
October  2014, 10(4): 1031-1039. doi: 10.3934/jimo.2014.10.1031

Bounds for the greatest eigenvalue of positive tensors

Zhen Wang 1, and  Wei Wu 1,

1. 

Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China, China

Received  October 2012 Revised  August 2013 Published  February 2014

Higher order tensors are generalizations of matrices. In this paper, we extend the bounds for the greatest eigenvalue of positive square matrices to positive tensors, and give further results on the bounds for the greatest eigenvector of positive tensors.
Keywords: positive tensor, Perron-Frobenius theorem, greatest eigenvector., greatest eigenvalue, Irreducible nonnegative tensor.
Mathematics Subject Classification: 15A18, 15A69, 15A7.
Citation: Zhen Wang, Wei Wu. Bounds for the greatest eigenvalue of positive tensors. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1031-1039. doi: 10.3934/jimo.2014.10.1031
References:
[1]

S. R. Bulò and M. Pelillo, A generalization of the Motzkin-Straus theorem to hypergraphs,, Optim. Lett., 3 (2009), 187.  doi: 10.1007/s11590-008-0108-3.  Google Scholar

[2]

S. R. Bulò and M. Pelillo, New bounds on the clique number of graphs based on spectral hypergraph theory,, in Learning and Intelligent Optimization, (2009), 45.   Google Scholar

[3]

K. C. Chang, K. Pearson and T. Zhang, On eigenvalue problems of real symmetric tensors,, J. Math. Anal. Appl., 350 (2009), 416.  doi: 10.1016/j.jmaa.2008.09.067.  Google Scholar

[4]

K. C. Chang, K. Pearson and T. Zhang, Perron Frobenius theorem for nonnegative tensors,, Comm. Math. Sci., 6 (2008), 507.  doi: 10.4310/CMS.2008.v6.n2.a12.  Google Scholar

[5]

L. Collatz, Einschliessungssatz die charakteristischen Zahlen von Matrizen,, Math. Zeit., 48 (1942), 221.  doi: 10.1007/BF01180013.  Google Scholar

[6]

J. Cooper and A. Dutle, Spectra of uniform hypergraphs,, Department of Mathematics, (2011).  doi: 10.1016/j.laa.2011.11.018.  Google Scholar

[7]

P. Drineas and L. H. Lim, A Multilinear Spectral Theory of Hyper-Graphs and Expander Hypergraphs,, 2005., ().   Google Scholar

[8]

S. Friedland, S. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions,, to appear in Linear Algebra and Its Applications, 438 (2013), 738.  doi: 10.1016/j.laa.2011.02.042.  Google Scholar

[9]

S. Hu and L. Qi, Algebraic connectivity of an even uniform hypergraph,, Journal of Combinatorial Optimization, 24 (2012), 564.  doi: 10.1007/s10878-011-9407-1.  Google Scholar

[10]

W. Li and M. Ng, Existence and Uniqueness of Stationary Probability Vector of a Transition Probability Tensor,, Technical report, (2011).   Google Scholar

[11]

L. H. Lim, Multilinear Pagerank: Measuring Higher Order Connectivity in Linked Objects,, The internet: Today and Tomorrow, (2005).   Google Scholar

[12]

L. H. Lim, Singular values and eigenvalues of tensors: A variational approach,, in Proceedings of the First IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), (2005), 129.   Google Scholar

[13]

Y. Liu, G. Zhou and N. F. Ibrahim, An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor,, Journal of Computational and Applied Mathematics, 235 (2010), 286.  doi: 10.1016/j.cam.2010.06.002.  Google Scholar

[14]

H. Minc, Nonnegative Matrices,, New York: John Wiley and Sons, (1988).   Google Scholar

[15]

M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor,, SIAM J. Matrix Anal. Appl., 31 (2009), 1090.  doi: 10.1137/09074838X.  Google Scholar

[16]

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

[17]

L. Qi, Eigenvalue and invariants of a tensor,, J. Math. Anal. Appl., 325 (2007), 1363.  doi: 10.1016/j.jmaa.2006.02.071.  Google Scholar

[18]

L. Qi, W. Sun and Y. Wang, Numerical multilinear algebra and its applications,, Front. Math. China, 2 (2007), 501.  doi: 10.1007/s11464-007-0031-4.  Google Scholar

[19]

Y. N. Yang and Q. Z. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors,, SIAM. J. Matrix Anal. Appl., 31 (2010), 2517.  doi: 10.1137/090778766.  Google Scholar

[20]

Y. N. Yang, Q. Z. Yang and Y. G. Li, An algorithm to find the spectral radius of nonnegative tensors and its convergence analysis,, , (2011).   Google Scholar

[21]

F. X. Zhang, The smoothing method for finding the largest eigenvalue of nonnegative matrices,, Numerical Mathematics: A Journal of Chinese Universities, 23 (2001), 45.   Google Scholar

[22]

L. Zhang and L. Qi, Linear convergence of an algorithm for computing the largest eigenvalue of a nonnegative tensor,, Numercal Linear Algebra with Applications, 19 (2012), 830.  doi: 10.1002/nla.822.  Google Scholar

show all references

References:
[1]

S. R. Bulò and M. Pelillo, A generalization of the Motzkin-Straus theorem to hypergraphs,, Optim. Lett., 3 (2009), 187.  doi: 10.1007/s11590-008-0108-3.  Google Scholar

[2]

S. R. Bulò and M. Pelillo, New bounds on the clique number of graphs based on spectral hypergraph theory,, in Learning and Intelligent Optimization, (2009), 45.   Google Scholar

[3]

K. C. Chang, K. Pearson and T. Zhang, On eigenvalue problems of real symmetric tensors,, J. Math. Anal. Appl., 350 (2009), 416.  doi: 10.1016/j.jmaa.2008.09.067.  Google Scholar

[4]

K. C. Chang, K. Pearson and T. Zhang, Perron Frobenius theorem for nonnegative tensors,, Comm. Math. Sci., 6 (2008), 507.  doi: 10.4310/CMS.2008.v6.n2.a12.  Google Scholar

[5]

L. Collatz, Einschliessungssatz die charakteristischen Zahlen von Matrizen,, Math. Zeit., 48 (1942), 221.  doi: 10.1007/BF01180013.  Google Scholar

[6]

J. Cooper and A. Dutle, Spectra of uniform hypergraphs,, Department of Mathematics, (2011).  doi: 10.1016/j.laa.2011.11.018.  Google Scholar

[7]

P. Drineas and L. H. Lim, A Multilinear Spectral Theory of Hyper-Graphs and Expander Hypergraphs,, 2005., ().   Google Scholar

[8]

S. Friedland, S. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions,, to appear in Linear Algebra and Its Applications, 438 (2013), 738.  doi: 10.1016/j.laa.2011.02.042.  Google Scholar

[9]

S. Hu and L. Qi, Algebraic connectivity of an even uniform hypergraph,, Journal of Combinatorial Optimization, 24 (2012), 564.  doi: 10.1007/s10878-011-9407-1.  Google Scholar

[10]

W. Li and M. Ng, Existence and Uniqueness of Stationary Probability Vector of a Transition Probability Tensor,, Technical report, (2011).   Google Scholar

[11]

L. H. Lim, Multilinear Pagerank: Measuring Higher Order Connectivity in Linked Objects,, The internet: Today and Tomorrow, (2005).   Google Scholar

[12]

L. H. Lim, Singular values and eigenvalues of tensors: A variational approach,, in Proceedings of the First IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), (2005), 129.   Google Scholar

[13]

Y. Liu, G. Zhou and N. F. Ibrahim, An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor,, Journal of Computational and Applied Mathematics, 235 (2010), 286.  doi: 10.1016/j.cam.2010.06.002.  Google Scholar

[14]

H. Minc, Nonnegative Matrices,, New York: John Wiley and Sons, (1988).   Google Scholar

[15]

M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor,, SIAM J. Matrix Anal. Appl., 31 (2009), 1090.  doi: 10.1137/09074838X.  Google Scholar

[16]

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

[17]

L. Qi, Eigenvalue and invariants of a tensor,, J. Math. Anal. Appl., 325 (2007), 1363.  doi: 10.1016/j.jmaa.2006.02.071.  Google Scholar

[18]

L. Qi, W. Sun and Y. Wang, Numerical multilinear algebra and its applications,, Front. Math. China, 2 (2007), 501.  doi: 10.1007/s11464-007-0031-4.  Google Scholar

[19]

Y. N. Yang and Q. Z. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors,, SIAM. J. Matrix Anal. Appl., 31 (2010), 2517.  doi: 10.1137/090778766.  Google Scholar

[20]

Y. N. Yang, Q. Z. Yang and Y. G. Li, An algorithm to find the spectral radius of nonnegative tensors and its convergence analysis,, , (2011).   Google Scholar

[21]

F. X. Zhang, The smoothing method for finding the largest eigenvalue of nonnegative matrices,, Numerical Mathematics: A Journal of Chinese Universities, 23 (2001), 45.   Google Scholar

[22]

L. Zhang and L. Qi, Linear convergence of an algorithm for computing the largest eigenvalue of a nonnegative tensor,, Numercal Linear Algebra with Applications, 19 (2012), 830.  doi: 10.1002/nla.822.  Google Scholar

[1]

Stefan Klus, Péter Koltai, Christof Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (1) : 51-79. doi: 10.3934/jcd.2016003
[2]

Stefan Klus, Christof Schütte. Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (2) : 139-161. doi: 10.3934/jcd.2016007
[3]

Martin Lustig, Caglar Uyanik. Perron-Frobenius theory and frequency convergence for reducible substitutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 355-385. doi: 10.3934/dcds.2017015
[4]

Gary Froyland, Ognjen Stancevic. Escape rates and Perron-Frobenius operators: Open and closed dynamical systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 457-472. doi: 10.3934/dcdsb.2010.14.457
[5]

Marianne Akian, Stéphane Gaubert, Antoine Hochart. A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 207-231. doi: 10.3934/dcds.2020009
[6]

Wen Li, Wei-Hui Liu, Seak Weng Vong. Perron vector analysis for irreducible nonnegative tensors and its applications. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-22. doi: 10.3934/jimo.2019097
[7]

King Hann Lim, Hong Hui Tan, Hendra G. Harno. Approximate greatest descent in neural network optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 327-336. doi: 10.3934/naco.2018021
[8]

Venkateswaran P. Krishnan, Plamen Stefanov. A support theorem for the geodesic ray transform of symmetric tensor fields. Inverse Problems & Imaging, 2009, 3 (3) : 453-464. doi: 10.3934/ipi.2009.3.453
[9]

Marc Kesseböhmer, Sabrina Kombrink. A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 335-352. doi: 10.3934/dcdss.2017016
[10]

Jan Boman, Vladimir Sharafutdinov. Stability estimates in tensor tomography. Inverse Problems & Imaging, 2018, 12 (5) : 1245-1262. doi: 10.3934/ipi.2018052
[11]

Mengmeng Zheng, Ying Zhang, Zheng-Hai Huang. Global error bounds for the tensor complementarity problem with a P-tensor. Journal of Industrial & Management Optimization, 2019, 15 (2) : 933-946. doi: 10.3934/jimo.2018078
[12]

Yiju Wang, Guanglu Zhou, Louis Caccetta. Nonsingular $H$-tensor and its criteria. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1173-1186. doi: 10.3934/jimo.2016.12.1173
[13]

M. S. Lee, B. S. Goh, H. G. Harno, K. H. Lim. On a two-phase approximate greatest descent method for nonlinear optimization with equality constraints. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 315-326. doi: 10.3934/naco.2018020
[14]

Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617
[15]

Nicolas Van Goethem. The Frank tensor as a boundary condition in intrinsic linearized elasticity. Journal of Geometric Mechanics, 2016, 8 (4) : 391-411. doi: 10.3934/jgm.2016013
[16]

Henry O. Jacobs, Hiroaki Yoshimura. Tensor products of Dirac structures and interconnection in Lagrangian mechanics. Journal of Geometric Mechanics, 2014, 6 (1) : 67-98. doi: 10.3934/jgm.2014.6.67
[17]

François Monard. Efficient tensor tomography in fan-beam coordinates. Inverse Problems & Imaging, 2016, 10 (2) : 433-459. doi: 10.3934/ipi.2016007
[18]

Kaili Zhang, Haibin Chen, Pengfei Zhao. A potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 429-443. doi: 10.3934/jimo.2018049
[19]

Nur Fadhilah Ibrahim. An algorithm for the largest eigenvalue of nonhomogeneous nonnegative polynomials. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 75-91. doi: 10.3934/naco.2014.4.75
[20]

Jiu Ding, Noah H. Rhee. A unified maximum entropy method via spline functions for Frobenius-Perron operators. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 235-245. doi: 10.3934/naco.2013.3.235

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences