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 (𝓖 1 , … , 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.

Abstract licence: CC BY-NC-ND

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