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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), 187-295. doi: 10.1007/s11590-008-0108-3.  Google Scholar

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S. R. Bulò and M. Pelillo, New bounds on the clique number of graphs based on spectral hypergraph theory, in Learning and Intelligent Optimization, (ed. T. Stützle), Springer Verlag, Berlin, (2009), 45-48. Google Scholar

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K. C. Chang, K. Pearson and T. Zhang, On eigenvalue problems of real symmetric tensors, J. Math. Anal. Appl., 350 (2009), 416-422. 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), 507-520. doi: 10.4310/CMS.2008.v6.n2.a12.  Google Scholar

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L. Collatz, Einschliessungssatz die charakteristischen Zahlen von Matrizen, Math. Zeit., 48 (1942), 221-226. doi: 10.1007/BF01180013.  Google Scholar

[6]

J. Cooper and A. Dutle, Spectra of uniform hypergraphs, Department of Mathematics, University of South Carolina, June 2011, Arxiv preprint arXiv:1106.4856, (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), 738-749. 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), 564-579. 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, Department of Mathematics, The Hong Kong Baptist University, 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), 129-132. 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), 286-292. doi: 10.1016/j.cam.2010.06.002.  Google Scholar

[14]

H. Minc, Nonnegative Matrices, New York: John Wiley and Sons, Inc, 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), 1090-1099. doi: 10.1137/09074838X.  Google Scholar

[16]

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

[17]

L. Qi, Eigenvalue and invariants of a tensor, J. Math. Anal. Appl., 325 (2007), 1363-1377. 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), 501-526. 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), 2517-2530. 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, arXiv:1102.2668, (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), 45-55, Chinese Series.  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), 830-841. 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), 187-295. 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, (ed. T. Stützle), Springer Verlag, Berlin, (2009), 45-48. Google Scholar

[3]

K. C. Chang, K. Pearson and T. Zhang, On eigenvalue problems of real symmetric tensors, J. Math. Anal. Appl., 350 (2009), 416-422. 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), 507-520. doi: 10.4310/CMS.2008.v6.n2.a12.  Google Scholar

[5]

L. Collatz, Einschliessungssatz die charakteristischen Zahlen von Matrizen, Math. Zeit., 48 (1942), 221-226. doi: 10.1007/BF01180013.  Google Scholar

[6]

J. Cooper and A. Dutle, Spectra of uniform hypergraphs, Department of Mathematics, University of South Carolina, June 2011, Arxiv preprint arXiv:1106.4856, (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), 738-749. 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), 564-579. 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, Department of Mathematics, The Hong Kong Baptist University, 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), 129-132. 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), 286-292. doi: 10.1016/j.cam.2010.06.002.  Google Scholar

[14]

H. Minc, Nonnegative Matrices, New York: John Wiley and Sons, Inc, 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), 1090-1099. doi: 10.1137/09074838X.  Google Scholar

[16]

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

[17]

L. Qi, Eigenvalue and invariants of a tensor, J. Math. Anal. Appl., 325 (2007), 1363-1377. 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), 501-526. 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), 2517-2530. 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, arXiv:1102.2668, (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), 45-55, Chinese Series.  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), 830-841. doi: 10.1002/nla.822.  Google Scholar

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