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

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. $\lambda \in R$ is called a ${\mathcal{B}}_r$-eigenvalue of $\mathcal{A}$ if $\mathcal{A}{x}^{m-1}=\lambda \mathcal{B}{x}^{m-1}$ for some $x\in R^n\backslash \{0\}$. 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.

24 pages

References in corpus (1)

• Perron-Frobenius theorem for nonnegative multilinear forms and extensions