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

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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

Abstract
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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 and 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.
[2]	K. C. Chang, K. Pearson and T. Zhang, Perron Frobenius theorem for nonnegative tensors, Commu Math Sci, 6 (2008), 507-520.
[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.
[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.
[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.
[6]	L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007.
[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.
[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.
[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.

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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.
[2]	K. C. Chang, K. Pearson and T. Zhang, Perron Frobenius theorem for nonnegative tensors, Commu Math Sci, 6 (2008), 507-520.
[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.
[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.
[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.
[6]	L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007.
[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.
[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.
[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.

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