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.

Keywords:

• tensor rank
• tensor decomposition
• multi-linear algebra
• singular value decomposition
• generalized Schur decomposition

AMS Subject Classifications:

• 15A03
• 15A59

Acknowledgements

The author thanks Charles Van Loan at Cornell University for the original insight into this problem, Carter Lyons and Stephen Lucas at James Madison University for thorough reading and writing suggestions, and the anonymous referee for the many suggestions to shorten and clarify this article. This research is partially funded by NSF grant DMS-0914974.