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

Lixing Han

doi:10.3934/naco.2013.3.583

Article Contents

2013, Volume 3, Issue 3: 583-599. Doi: 10.3934/naco.2013.3.583

This issue Previous Article Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities Next Article

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

Lixing Han^1,

1.
Department of Mathematics, University of Michigan-Flint, Flint, MI 48502

Received: June 2012

Revised: April 2013

Published: June 2013

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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. $\lambda \in \mathbb{R}$ is called a ${\mathcal B}_r$-eigenvalue of ${\mathcal A}$ if ${\mathcal A} x^{m-1} = \lambda {\mathcal B} x^{m-1}$ for some $x \in \mathbb{R}^n \backslash \{0\}$. 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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Mathematics Subject Classification: Primary: 65F15, 65K05; Secondary: 15A69.

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