ALAMT  Vol.5 No.1 , March 2015
H-Singular Value of a Positive Tensor
Author(s) Jun He*
ABSTRACT

In this paper we study properties of H-singular values of a positive tensor and present an iterative algorithm for computing the largest H-singular value of the positive tensor. We prove that this method converges for any positive tensors.


Cite this paper
He, J. (2015) H-Singular Value of a Positive Tensor. Advances in Linear Algebra & Matrix Theory, 5, 16-24. doi: 10.4236/alamt.2015.51002.
References
[1]   Chang, K.-C., Pearson, K. and Zhang, T. (2011) Primitivity, the Convergence of the NZQ Method, and the Largest Eigenvalue for Nonnegative Tensors. SIAM Journal on Matrix Analysis and Applications, 32, 806-819.
http://dx.doi.org/10.1137/100807120

[2]   Qi, L.Q. (2007) Eigenvalues and Invariants of Tensor. Journal of Mathematical Analysis and Applications, 325, 1363-1377.http://dx.doi.org/10.1016/j.jmaa.2006.02.071

[3]   Pearson, K.J. (2010) Primitive Tensors and Convergence of an Iterative Process for the Eigenvalue of a Primitive Tensor. http://arxiv.org/abs/1004.2423

[4]   Wang, Y.J., Qi, L.Q. and Zhang, X.Z. (2009) A Practical Method for Computing the Largest M-Eigenvalue of a Fourth-Order Partially Symmetric Tensor. Numerical Linear Algebra with Applications, 16, 589-601.
http://dx.doi.org/10.1002/nla.633

[5]   Qi, L.Q., Wang, Y.J. and Wu, E.X. (2008) D-Eigenvalues of Diffusion Kurtosis Tensor. Journal of Computational and Applied Mathematics, 221, 150-157. http://dx.doi.org/10.1016/j.cam.2007.10.012

[6]   Ng, M., Qi, L.Q. and Zhou, G.L. (2009) Finding the Largest Eigenvalue of a Non-Negative Tensor. SIAM Journal on Matrix Analysis and Applications, 31, 1090-1099. http://dx.doi.org/10.1137/09074838X

[7]   Chang, K.-C., Pearson, K. and Zhang, T. (2008) Perron-Frobenius Theorem for Nonnegative Tensors. Communications in Mathematical Sciences, 6, 507-520. http://dx.doi.org/10.4310/CMS.2008.v6.n2.a12

[8]   Chang, K.-C., Pearson, K. and Zhang, T. (2009) On Eigenvalue Problems of Real Symmetric Tensors. Journal of Mathematical Analysis and Applications, 350, 416-422.
http://dx.doi.org/10.1016/j.jmaa.2008.09.067

[9]   Qi, L.Q. (2005) Eigenvalues of a Real Supersymmetric Tensor. Journal of Symbolic Computation, 40, 1302-1324.
http://dx.doi.org/10.1016/j.jsc.2005.05.007

[10]   Liu, Y.J., Zhou, G.L. and Ibrahim, N.F. (2010) An Always Convergent Algorithm for the Largest Eigenvalue of an Irreducible Nonnegative Tensor. Journal of Computational and Applied Mathematics, 235, 286-292.
http://dx.doi.org/10.1016/j.cam.2010.06.002

[11]   Chang, K.C., Qi, L.Q. and Zhou, G.L. (2010) Singular Values of a Real Rectangular Tensor. Journal of Mathematical Analysis and Applications, 370, 284-294. http://dx.doi.org/10.1016/j.jmaa.2010.04.037

[12]   Yang, Y. and Yang, Q. (2011) Singular Values of Nonnegative Rectangular Tensors. Frontiers of Mathematics in China, 6, 363-378.

[13]   Zhou, G.K., Caccetta, L. and Qi, L.Q. (2013) Convergence of an Algorithm for the Largest Singular Value of a Nonnegative Rectangular Tensor. Linear Algebra and Its Applications, 438, 959-968.
http://dx.doi.org/10.1016/j.laa.2011.06.038

[14]   Lim, L.H. (2005) Singular Values and Eigenvalues of Tensors: A Variational Approach. Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP’05), 1, 129-132.

[15]   Yang, Y.N. and Yang, Q.Z. (2010) Further Results for Perron-Frobenius Theorem for Nonnegative Tensors. SIAM Journal on Matrix Analysis and Applications, 31, 2517-2530.
http://dx.doi.org/10.1137/090778766

 
 
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