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It is known that a best low-rank approximation to multi-way arrays orhigher-order tensors may not exist. This is due to the fact that the set ofmulti-way arrays with rank at most $R$ is not closed. Nonexistence of the bestlow-rank approximation results in diverging rank-1 components when an attemptis made to compute the approximation. Recently, a solution to this problem hasbeen proposed for real $I\times J\times 2$ arrays. Instead of a best rank-$R$approximation the best fitting Generalized Schur Decomposition (GSD) iscomputed. Under the restriction of nonsingular upper triangular matrices in theGSD, the set of GSD solutions equals the interior and boundary of the rank-$R$set. Here, we show that this holds even without the restriction. We provide acomplete classification of interior, boundary, and exterior points of therank-$R$ set of real $I\times J\times 2$ arrays, and show that the set of GSDsolutions equals the interior and boundary of this set.

The Generalized Schur Decomposition and the rank-$R$ set of real $I\times J\times 2$ arrays