Maximal number of distinct H-eigenpairs for a two-dimensional real tensor

Abstract

Based on the generalized characteristic polynomial introduced by J. Canny in Generalized characteristic polynomials [J. Symbolic Comput., 1990, 9(3): 241–250], it is immediate that for any m-order n-dimensional real tensor, the number of distinct H-eigenvalues is less than or equal to n(m−1) ⁿ⁻¹. However, there is no known bounds on the maximal number of distinct Heigenvectors in general. We prove that for any m ⩾ 2, an m-order 2-dimensional tensor A exists such that A has 2(m − 1) distinct H-eigenpairs. We give examples of 4-order 2-dimensional tensors with six distinct H-eigenvalues as well as six distinct H-eigenvectors. We demonstrate the structure of eigenpairs for a higher order tensor is far more complicated than that of a matrix. Furthermore, we introduce a new class of weakly symmetric tensors, called p-symmetric tensors, and show under certain conditions, p-symmetry will effectively reduce the maximal number of distinct H-eigenvectors for a given two-dimensional tensor. Lastly, we provide a complete classification of the H-eigenvectors of a given 4-order 2-dimensional nonnegative p-symmetric tensor. Additionally, we give sufficient conditions which prevent a given 4-order 2-dimensional nonnegative irreducible weakly symmetric tensor from possessing six pairwise distinct H-eigenvectors.

Keywords

Symmetric tensor / H-eigenpairs

References

[1] Canny J.. Generalized characteristic polynomials. J Symbolic Comput, 1990, 9(3): 241-250

[2] Chang K. C., Pearson K., Zhang T.. On eigenvalue problems of real symmetric tensors. J Math Anal Appl, 2009, 350: 416-422

[3] Chang K. C., Pearson K., Zhang T.. Perron-Frobenius theorem for nonnegative tensors. Commun Math Sci, 2008, 6(2): 507-520

[4] Chang K. C., Pearson K., Zhang T.. Primitivity, the convergence of the NQZ method, and the largest eigenvalue for nonnegative tensors. SIAM J Matrix Anal Appl, 2011, 32: 806-819

[5] Cartwright D, Sturmfels B. The number of eigenvalues of a tensor. Linear Algebra Appl (to appear)

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

[7] Friedland S, Gaubert S, Han L. Perron-Frobenius theorem for nonnegative multilinear forms and extensions. Linear Algebra Appl (to appear)

[8] Gaubert S., Gunawardena J.. The Perron-Frobenius theorem for homogeneous, monotone functions. Trans Amer Math Soc, 2004, 356(12): 4931-4950

[9] Konvalina J., Matache V.. Palindrome-polynomials with roots on the unit circle. C R Math Acad Sci Soc R Can, 2004, 26(2): 39-44

[10] Lim L H. Singular values and eigenvalues of tensors, A variational approach. Proc 1st IEEE International Workshop on Computational Advances of Multi-tensor Adaptive Processing, Dec 13–15, 2005. 2005, 129–132

[11] Lim L H. Multilinear pagerank: measuring higher order connectivity in linked objects. The Internet: Today and Tomorrow, July, 2005

[12] Liu Y., Zhou G., Ibrahim N. F.. An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor. J Comput Appl Math, 2010, 235(1): 286-292

[13] Ng M., Qi L., Zhou G.. Finding the largest eigenvalue of a nonnegative tensor. SIAM J Matirx Anal Appl, 2010, 31(3): 1090-1099

[14] Markovsky I, Rao S. Palindromic polynomials, time-reversible systems, and conserved quantities. In: 16th Mediterranean Conference on Control and Automation, Congress Centre, Ajaccio, France, June 25–27, 2008

[15] Qi L.. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302-1324

[16] Qi L.. Eigenvalues and invariants of tensors. J Math Anal Appl, 2007, 325: 1363-1377

[17] Yang Y., Yang Q.. Further results for Perron-Frobenius Theorem for nonnegative tensors. SIAM J Matrix Anal Appl, 2010, 31(5): 2517-2530

[18] Zhang T.. Existence of real eigenvalues of real tensors. Nonlinear Anal, 2011, 74: 2862-2868