Tensor Decompositions with applications to LU and SLOCC equivalence of multipartite pure states

We introduce a general lemma, of which one consequence is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus' (2010) algorithm used the HOSVD to compute normal forms of generic n-qubit pure states under the action of the local unitary group U₂× n. Taking advantage of the double cover SL₂(C) × SL₂(C) → SO₄(C), we produce similar algorithms (distinguished by the parity of n) that compute normal forms for tensors in (C²)⊗ n for the action of the SLOCC group SL₂(C)× n.