RESEARCH ARTICLE

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

  • Kelly J. PEARSON ,
  • Tan ZHANG
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  • Department of Mathematics and Statistics, Murray State University, Murray, KY 42071-0009, USA

Received date: 28 Mar 2012

Accepted date: 27 Oct 2012

Published date: 01 Feb 2013

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

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.

Cite this article

Kelly J. PEARSON , Tan ZHANG . Maximal number of distinct H-eigenpairs for a two-dimensional real tensor[J]. Frontiers of Mathematics in China, 2013 , 8(1) : 85 -105 . DOI: 10.1007/s11464-012-0263-9

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