A note on the three-way generalization of the Jordan canonical form

The limit point 𝓧 of an approximating rank-R sequence of a tensor Ƶ can be obtained by fitting a decomposition (S, T, U) ⋅ 𝓖 to Ƶ. The decomposition of the limit point 𝓧 = (S, T, U) ⋅ 𝓖 with 𝓖 = blockdiag(𝓖1, … , 𝓖m) can be seen as a three order generalization of the real Jordan canonical form. The main aim of this paper is to study under what conditions we can turn 𝓖j into canonical form if some of the upper triangular entries of the last three slices of 𝓖j are zeros. In addition, we show how to turn 𝓖j into canonical form under these conditions.