An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors
For researchers in tensor computation, this provides a new optimization-based method for eigenvalue problems, though it is an incremental extension of existing matrix principles.
The paper extends Auchmuty's unconstrained variational principles for eigenvalues of symmetric matrices to even order symmetric tensors, introducing two unconstrained optimization problems to find real eigenvalues. Numerical results demonstrate effectiveness for finding Z-eigenvalues and determining positive semidefiniteness.
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. $λ\in R$ is called a ${\mathcal B}_r$-eigenvalue of ${\mathcal A}$ if ${\mathcal A} x^{m-1} = λ{\mathcal B} x^{m-1}$ for some $x \in R^n \backslash \{0\}$. 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.