Frontiers of Mathematics in China >

2011 , Vol. 6 >Issue 2: 363 - 378

DOI: https://doi.org/10.1007/s11464-011-0108-y

RESEARCH ARTICLE

Singular values of nonnegative rectangular tensors

  • Yuning YANG ,
  • Qingzhi YANG
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  • School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Received date: 19 May 2010

Accepted date: 08 Jan 2011

Published date: 01 Apr 2011

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
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Abstract

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. Some properties concerning the singular values of a real rectangular tensor were discussed by K. C. Chang et al. [J. Math. Anal. Appl., 2010, 370: 284-294]. In this paper, we give some new results on the Perron-Frobenius Theorem for nonnegative rectangular tensors. We show that the weak Perron-Frobenius keeps valid and the largest singular value is really geometrically simple under some conditions. In addition, we establish the convergence of an algorithm proposed by K. C. Chang et al. for finding the largest singular value of nonnegative primitive rectangular tensors.

Key words： Nonnegative rectangular tensor; Perron-Frobenius Theorem; singular value; algorithm

Cite this article

Yuning YANG , Qingzhi YANG . Singular values of nonnegative rectangular tensors[J]. Frontiers of Mathematics in China, 2011 , 6(2) : 363 -378 . DOI: 10.1007/s11464-011-0108-y

References

Publishing order | Descend order by publishing year | Descend order by cited within
1
Bloy L, Verma R. On computing the underlying fiber directions from the diffusion orientation distribution function. In: Metaxas D, Axel L, Fichtinger G, Székely G, eds. Medical Image Computing and Computer-Assisted Intervention-MICCAI 2008. Lecture Notes in Computer Science, No 5241. Berlin/Heidelberg: Springer, 2008, 1-8

DOI

2
Chang K C, Pearson K, T. Zhang T. Perron Frobenius Theorem for nonnegative tensors. Comm Math Sci, 2008, 6: 507-520

3
Chang K C, Qi L, Zhou G. Singular values of a real rectangular tensor. J Math Anal Appl, 2010, 370: 284-294

DOI

4
Drineas P, Lim L H. A multilinear spectral theory of hypergraphs and expander hypergraphs. 2005

5
Lathauwer L D, Moor B D, Vandewalle J. On the best rank-1 and rank-(R1, R2, . . . , RN) approximation of higher-order tensors. SIAM J Matrix Anal Appl, 2000, 21: 1324-1342

DOI

6
Lim L H. Singular values and eigenvalues of tensors: a variational approach. Proceedings of the IEEE InternationalWorkshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005, 1: 129-132

7
Lim L H. Multilinear pagerank: measuring higher order connectivity in linked objects. The Internet: Today and Tomorrow, <month>July</month>, 2005

8
Ng M, Qi L, Zhou G. Finding the largest eigenvalue of a non-negative tensor. SIAM J Matrix Anal Appl, 2009, 31: 1090-1099

DOI

9
Ni Q, Qi L, Wang F. An eigenvalue method for the positive definiteness identification problem. IEEE Transactions on Automatic Control, 2008, 53: 1096-1107

DOI

10
Pearson K. Essentially positive tensors. Int J Algebra, 2010, 4: 421-427

11
Pearson K. Primitive tensors and convergence of an iterative process for the eigenvalues of a primitive tensor. arXiv: 1004.2423v1, 2010

12
Qi L. Eigenvalues of a real supersymmetric tensor. J Symb Comput, 2005, 40: 1302-1324

DOI

13
Qi L, Sun W, Wang Y. Numerical multilinear algebra and its applications. Front Math China, 2007, 2(4): 501-526

DOI

14
Qi L, Wang Y, Wu E. D-eigenvalues of diffusion kurtosis tensor. J Comput Appl Math, 2008, 221: 150-157

DOI

15
Yang Y, Yang Q. Further results for Perron-Frobenius Theorem for nonnegative tensors. SIAM J Matrix Anal Appl, 2010, 31: 2517-2530

DOI

16
Yang Y, Yang Q. A note on the geometric simplicity of the spectral radius of nonnegative irreducible tensor. http://arxiv.org/abs/1101.2479v1, 2010

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