Jordan canonical form of three-way tensor with multilinear rank (4,4,3)

Abstract

The Jordan canonical form of matrix is an important concept in linear algebra. And the concept has been extended to three-way arrays case. In this paper, we study the Jordan canonical form of three-way tensor with multilinear rank (4,4,3). For a 4×4×4 tensor {\mathfrak{G}}_j with multilinear rank (4,4,3), we show that {\mathfrak{G}}_j must be turned into the canonical form if the upper triangular entries of the last three slices of {\mathfrak{G}}_j are nonzero. If some of the upper triangular entries of the last three slices of {\mathfrak{G}}_j are zeros, we give some conditions to guarantee that {\mathfrak{G}}_j can be turned into the canonical form.

Key words： Jordan canonical form; tensor decomposition; multilinear rank

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