All Real Eigenvalues of Symmetric Tensors

Abstract

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.

SDP relaxation method for detecting P-tensors

“…Later shifted power methods are proposed in [8,19]. Recently, SDP relaxation method is applied to compute Z(H)-eigenvalues in [1].…”

“…and the sequences {ρ (1) k } and {ρ (2) k } are monotonically increasing. Furthermore, suppose y * is a minimizer of (17).…”

“…Tensor A is a P-tensor if ρ (1) k > 0, and tensor A is a P 0 -tensor but not a P-tensor if and only if f s = 0. Furthermore, tensor A is not a P 0 -tensor if and only if f s < 0.…”

“…k > 0, then A is a P-tensor and stop; if ρ (1) k = 0 and rank condition (19) with y * k holds for some t, then A is a P 0 -tensor but not P-tensor and stop; and if ρ (1) k < 0, rank condition (19) is satisfied for some t, then A is not a P 0 -tensor and stop. If rank condition (19) with y * k fails, let k := k + 1 and go to Step 1.…”

“…The software Gloptipoly 3 [7] is used to solve the semidefinite relaxations. For convenience, we use the following notation: for any i 1 and S π(i1i2···im) to denote the set of all these permutations. ρ …”