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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.

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Let A ⊆ N n be a finite set, and K ⊆ R n be a compact semialgebraic set. An A-truncated multisequence (A-tms) is a vector y = (yα) indexed by elements in A. The A-truncated K-moment problem (A-TKMP) concerns whether or not a given A-tms y admits a K-measure µ, i.e., µ is a nonnegative Borel measure supported in K such that yα = K x α dµ for all α ∈ A. This paper proposes a numerical algorithm for solving A-TKMPs. It aims at finding a flat extension of y by solving a hierarchy of semidefinite relaxations {(SDR) k } ∞ k=1 for a moment optimization problem, whose objective R is generated in a certain randomized way. If y admits no K-measures and R[x] A is K-full (there exists p = α∈A pαx α that is positive on K), then (SDR) k is infeasible for all k big enough, which gives a certificate for the nonexistence of representing measures. If y admits a K-measure, then for almost all generated R, this algorithm has the following properties: i) we can asymptotically get a flat extension of y by solving the hierarchy {(SDR) k } ∞ k=1 ; ii) under a general condition that is almost sufficient and necessary, we can get a flat extension of y by solving (SDR) k for some k; iii) the obtained flat extensions admit a ratomic K-measure with r ≤ |A|. The decomposition problems for completely positive matrices and sums of even powers of real linear forms, and the standard truncated K-moment problems, are special cases of A-TKMPs. They can be solved numerically by this algorithm. R A (K) := {y ∈ R A : meas(y, K) = ∅}. 1991 Mathematics Subject Classification. 44A60, 47A57, 90C22, 90C90. Key words and phrases. A-truncated multisequence, A-truncated K-moment problem, completely positive matrices, flat extension, moment matrix, localizing matrix, representing measure, semidefinite program, sums of even powers. 1 For each J ⊆ {1, . . . , m 2 + 1}, denoteThe set of critical points of (A.1) with the active set J is contained in V J , which is called a critical variety of (A.1). We show that if the coefficients of f hom , h hom , g hom J satisfy some discriminantal inequalities, then V J is finite. We refer to [33, Section 3]) for the definition of discriminants ∆. Proposition A.1. Let f, h i (i ∈ [m 1 ]), g j (j ∈ [m 2 + 1]) ∈ R[x], and V J be defined as above. For any J ⊆ [m 2 + 1], if ∆(f hom , h hom , g hom J ) = 0, ∆(h hom , g hom J ) = 0, (A.3) rank Jac(f hom , h hom , g hom J )| x ≤ m 1 + |J|, h hom (x) = g hom J (x) = 0. Since ∆(h hom , g hom J ) = 0, rank Jac(h hom , g hom J )| x = m 1 + |J| for all 0 = x ∈ V C (h hom , g hom J ). The rank condition in (A.3) implies that f hom (v) = 0 (cf. [33, Section 3]). Hence, v is a nonzero singular solution to f hom (x) = h hom (x) = g hom J (x) = 0, which contradicts ∆(f hom , h hom , g hom J ) = 0 (cf. [33]). So, V J must be finite.