All Real Eigenvalues of Symmetric Tensors

This paper studies how to compute all real eigenvalues, associated to real eigenvectors, of a symmetric tensor. As is well known, the largest or smallest eigenvalue can be found by solving a polynomial optimization problem, while the other middle ones cannot. We propose a new approach for computing all real eigenvalues sequentially, from the largest to the smallest. It uses Jacobian semidefinite relaxations in polynomial optimization. We show that each eigenvalue can be computed by solving a finite hierarchy of semidefinite relaxations. Numerical experiments are presented to show how to do this.