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Maximal number of distinct H-eigenpairs for a two-dimensional real tensor |
Kelly J. PEARSON, Tan ZHANG() |
Department of Mathematics and Statistics, Murray State University, Murray, KY 42071-0009, USA |
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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.
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Keywords
Symmetric tensor
H-eigenpairs
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Corresponding Author(s):
ZHANG Tan,Email:tzhang@murraystate.edu
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Issue Date: 01 February 2013
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1 |
Canny J. Generalized characteristic polynomials. J Symbolic Comput , 1990, 9(3): 241-250 doi: 10.1016/S0747-7171(08)80012-0
|
2 |
Chang K C, Pearson K, Zhang T. On eigenvalue problems of real symmetric tensors. J Math Anal Appl , 2009, 350: 416-422 doi: 10.1016/j.jmaa.2008.09.067
|
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 doi: 10.1137/100807120
|
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 doi: 10.1090/S0002-9947-04-03470-1
|
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 doi: 10.1016/j.cam.2010.06.002
|
13 |
Ng M, Qi L, Zhou G. Finding the largest eigenvalue of a nonnegative tensor. SIAM J Matrix Anal Appl , 2009, 31(3): 1090-1099 doi: 10.1137/09074838X
|
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 , June25-27, 2008
|
15 |
Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput , 2005, 40: 1302-1324 doi: 10.1016/j.jsc.2005.05.007
|
16 |
Qi L. Eigenvalues and invariants of tensors. J Math Anal Appl , 2007, 325: 1363-1377 doi: 10.1016/j.jmaa.2006.02.071
|
17 |
Yang Y, Yang Q. Further results for Perron-Frobenius Theorem for nonnegative tensors. SIAM J Matrix Anal Appl , 2010, 31(5): 2517-2530 doi: 10.1137/090778766
|
18 |
Zhang T. Existence of real eigenvalues of real tensors. Nonlinear Anal , 2011, 74: 2862-2868 doi: 10.1016/j.na.2011.01.008
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