Inverse eigenvalue problem for tensors

Abstract

Given a tensor T ∈ T(C n , m + 1), the space of tensors of order m + 1 and dimension n with complex entries, it has nm n−1 eigenvalues (counted with algebraic multiplicities). The inverse eigenvalue problem for tensors is a generalization of that for matrices. Namely, given a multiset S ∈ C nm n−1 /S(nm n−1 ) of total multiplicity nm n−1 , is there a tensor in T(C n , m + 1) such that the multiset of eigenvalues of T is exactly S? The solvability of the inverse eigenvalue problem for tensors is studied in this article. With tools from algebraic geometry, it is proved that the necessary and sufficient condition for this inverse problem to be generically solvable is m = 1, or n = 2, or (n, m) = (3, 2), (4, 2), (3, 3).2010 Mathematics Subject Classification. 15A18; 15A69; 65F18.