A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization

Abstract

Canonical decomposition is a key concept in multilinear algebra. In this paper we consider the decomposition of higher‐order tensors which have the property that the rank is smaller than the greatest dimension. We derive a new and relatively weak deterministic sufficient condition for uniqueness. The proof is constructive. It shows that the canonical components can be obtained from a simultaneous matrix diagonalization by congruence, yielding a new algorithm. From the deterministic condition we derive an easy‐to‐check dimensionality condition that guarantees generic uniqueness.