Abstract
By the resultant theory, the E-characteristic polynomial of a real rectangular tensor is defined. It is proved that an E-singular value of a real rectangular tensor is always a root of the E-characteristic polynomial. The definition of the regularity of square tensors is generalized to the rectangular tensors, and in the regular case, a root of the Echaracteristic polynomial of a special rectangular tensor is an E-singular value of the rectangular tensor. Moreover, the best rank-one approximation of a real partially symmetric rectangular tensor is investigated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Qi L Q. Eigenvalues of a real supersymmetric tensor[J]. Journal of Symbolic Computation, 2005, 40(6): 1302–1324.
Lim L H. Singular values and eigenvalues of tensors: A variational approach[C]. In: Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing. Puerto Vallarta, Mexico, 2005.
Qi L Q, Wang Y J, Wu E X. D-eigenvalues of diffusion kurtosis tensors[J]. Journal of Computational and Applied Mathematics, 2008, 221(1): 150–157.
Ng M, Qi L Q, Zhou G L. Finding the largest eigenvalue of a nonnegative tensor[J]. SIAM Journal on Matrix Analysis and Applications, 2009, 31(3): 1090–1099.
Qi L Q, Wang F, Wang Y J. Z-eigenvalue methods for a global polynomial optimization problem[J]. Mathematical Programming, 2009, 118(2): 301–316.
De Lathauwer L, De Moor B, Vandewalle J. On the best rank-1 and rank-(R 1,R 2,...,R N) approximation of higherorder tensors[J]. SIAM Journal on Matrix Analysis and Applications, 2000, 21(4): 1324–1342.
Chang K, Qi L Q, Zhou G L. Singular values of a real rectangular tensor[J]. Journal of Mathematical Analysis and Applications, 2010, 370(1): 284–294.
Zhao N. E-singular values of a real partially symmetric rectangular tensor[J]. Journal of Shandong University( Natural Science), 2012, 47(10):34–37 (in Chinese).
Qi L Q. Eigenvalues and invariants of tensors[J]. Journal of Mathematical Analysis and Applications, 2007, 325(2):1363–1377.
Li A M, Qi L Q, Zhang B. E-characteristic polynomials of tensors[J]. Communications in Mathematical Sciences, 2013, 11(1): 33–53.
Hu S L, Qi L Q. The E-characteristic polynomial of a tensor of dimension 2[J]. Applied Mathematics Letters, 2013, 26(2): 225–231.
Qi L Q, Dai H H, Han D R. Conditions for strong ellipticity and M-eigenvalues[J]. Frontiers of Mathematics in China, 2009, 4(2): 349–364.
Ling C, Nie J W, Qi L Q et al. Biquadratic optimization over unit spheres and semidefinite programming relaxations[J]. SIAM Journal on Optimization, 2009, 20(3): 1286–1310.
Dahl G, Leinaas J M, Myrheim J et al. A tensor product matrix approximation problem in quantum physics[J]. Linear Algebra and Its Applications, 2007, 420(2/3): 711–725.
Wang Y J, Qi L Q, Zhang X Z. A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor[J]. Numerical Linear Algebra with Applications, 2009, 16(7): 589–601.
Cox D A, Little J, O’Shea D. Using Algebraic Geometry[M]. Springer-Verlag, New York, USA, 1998.
Author information
Authors and Affiliations
Corresponding author
Additional information
Wu Wei, born in 1969, female, Dr, associate Prof.
Rights and permissions
About this article
Cite this article
Wu, W., Chen, X. E-characteristic polynomials of real rectangular tensor. Trans. Tianjin Univ. 20, 232–235 (2014). https://doi.org/10.1007/s12209-014-2110-4
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12209-014-2110-4