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2011, 1(3): 381-388. doi: 10.3934/naco.2011.1.381

Some results on $l^k$-eigenvalues of tensor and related spectral radius

Chen Ling 1, and  Liqun Qi 2,

1. 

School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China
2. 

Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong

Received  April 2011 Revised  July 2011 Published  September 2011

In this paper, we study the $l^k$-eigenvalues/vectors of a real symmetric square tensor. Specially, we investigate some properties on the related $l^k$-spectral radius of a real nonnegative symmetric square tensor.
Keywords: $l^k$-eigenvalue, irreducibility., $l^k$-spectral radius, Nonnegative tensor.
Mathematics Subject Classification: Primary: 15A18, 15A69; Secondary: 90C3.
Citation: Chen Ling, Liqun Qi. Some results on $l^k$-eigenvalues of tensor and related spectral radius. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 381-388. doi: 10.3934/naco.2011.1.381
References:
[1]

L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, in "Medical Image Computing and Computer-Assisted Intervention MICCAI 2008'' (eds. Dimitris Metaxas, Leon Axel, Gabor Fichtinger and Gábor Székely), Springer-Verlag, Berlin/Heidelberg, (2008), 1-8. Google Scholar

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[3]

K. C. Chang, L. Qi and G. Zhou, Singular values of a real rectangular tensor, J. Math. Anal. Appl., 370 (2010), 284-294. doi: 10.1016/j.jmaa.2010.04.037.  Google Scholar

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M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a non-negative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099. doi: 10.1137/09074838X.  Google Scholar

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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

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L. Qi, F. Wang and Y. Wang, Z-eigenvalue methods for a global polynomial optimization problem, Math. Program., 118 (2009), 301-316. doi: 10.1007/s10107-007-0193-6.  Google Scholar

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Y. 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

show all references

References:
[1]

L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, in "Medical Image Computing and Computer-Assisted Intervention MICCAI 2008'' (eds. Dimitris Metaxas, Leon Axel, Gabor Fichtinger and Gábor Székely), Springer-Verlag, Berlin/Heidelberg, (2008), 1-8. Google Scholar

[2]

K. C. Chang, K. Pearson and T. Zhang, Perron Frobenius theorem for nonnegative tensors, Commu Math Sci, 6 (2008), 507-520.  Google Scholar

[3]

K. C. Chang, L. Qi and G. Zhou, Singular values of a real rectangular tensor, J. Math. Anal. Appl., 370 (2010), 284-294. doi: 10.1016/j.jmaa.2010.04.037.  Google Scholar

[4]

L. H. Lim, Singular values and eigenvalues of tensors: a variational approach, in "Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Addaptive Processing, CAMSAP'05'', 1, IEEE Computer Society Press, Piscataway, (2005), 129-132. Google Scholar

[5]

M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a non-negative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099. doi: 10.1137/09074838X.  Google Scholar

[6]

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

[7]

L. Qi, F. Wang and Y. Wang, Z-eigenvalue methods for a global polynomial optimization problem, Math. Program., 118 (2009), 301-316. doi: 10.1007/s10107-007-0193-6.  Google Scholar

[8]

L. Qi, G. Yu and E. X. Wu, Higher order positive semi-definite diffusion tensor imaging, SIAM J. Imaging Sci., 3 (2010), 416-433.  Google Scholar

[9]

Y. 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

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