The rank of a 2 × 2 × 2 tensor

Abstract

As computing power increases, many more problems in engineering and data analysis involve computation with tensors, or multi-way data arrays. Most applications involve computing a decomposition of a tensor into a linear combination of rank-1 tensors. Ideally, the decomposition involves a minimal number of terms, i.e. computation of the rank of the tensor. Tensor rank is not a straight-forward extension of matrix rank. A constructive proof based on an eigenvalue criterion is provided that shows when a 2 × 2 × 2 tensor over ℝ is rank-3 and when it is rank-2. The results are extended to show that n × n × 2 tensors over ℝ have maximum possible rank n + k where k is the number of complex conjugate eigenvalue pairs of the matrices forming the two faces of the tensor cube.