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 A be an mth order n-dimensional real weakly symmetric tensor and B be a real weakly symmetric positive definite tensor of the same size. λ∈ℝ is called a Br-eigenvalue of A if Axm-1=λBxm-1 for some x∈ℝn\{0}. In this paper, we introduce two unconstrained optimization problems and obtain some variational characterizations for the minimum and maximum Br-eigenvalues of 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.