An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors

2012/03/23 by Han, Lixing
#15A69 #65F15 #65K05 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1203.5150

Abstract

Let n be a positive integer and m be a positive even integer. Let \mathcal A be an m^th order n-dimensional real weakly symmetric tensor and \mathcal B be a real weakly symmetric positive definite tensor of the same size. λ∈ R is called a \mathcal B_r-eigenvalue of \mathcal A if \mathcal A x^m-1 = λ\mathcal B x^m-1 for some x ∈ Rⁿ \backslash \0\. In this paper, we introduce two unconstrained optimization problems and obtain some variational characterizations for the minimum and maximum \mathcal B_r--eigenvalues of \mathcal A. Our results extend Auchmuty's unconstrained variational principles for eigenvalues of real symmetric matrices. This unconstrained optimization approach can be used to find a Z-, H-, or D-eigenvalue of an even order weakly symmetric tensor. We provide some numerical results to illustrate the effectiveness of this approach for finding a Z-eigenvalue and for determining the positive semidefiniteness of an even order symmetric tensor.

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