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, … , 𝓖 m ) 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 𝓖 j into canonical form if some of the upper triangular entries of the last three slices of 𝓖 j are zeros. In addition, we show how to turn 𝓖 j into canonical form under these conditions.