We show that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a tensor possesses a given eigenvalue, singular value, or spectral norm; approximating an eigenvalue, eigenvector, singular vector, or spectral norm; determining a best rank-1 approximation to a tensor; and determining the rank of a tensor. Additionally, we prove that some of these problems have no polynomial time approximation schemes, some are undecidable over the rationals, and at least one enumerative version is #P-complete. We also show that restricting these problems to symmetric tensors does not alleviate their NP-hardness and that the problem of deciding nonnegative definiteness of a symmetric 4-tensor is also NP-hard. Except for this nonnegative definiteness problem, all our results apply to 3-tensors. We shall argue that these 3-tensor problems may be viewed as the boundary separating the computational tractability of linear/convex problems from the intractability of nonlinear/nonconvex ones.

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