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• W2019108549 abstract "Three-way arrays (or tensors) can be regarded as extensions of the traditional two-way data matrices that have a third dimension. Studying algebraic properties of arrays is relevant, for example, for the Tucker three-way PCA method, which generalizes principal component analysis to three-way data. One important algebraic property of arrays is concerned with the possibility of transformations to simplicity. An array is said to be transformed to a simple form when it can be manipulated by a sequence of invertible operations such that a vast majority of its entries become zero. This paper shows how 3 × 3 × 3 arrays, whether symmetric or nonsymmetric, can be transformed to a simple form with 18 out of its 27 entries equal to zero. We call this simple form the “knight's move pattern” due to a loose resemblance to the moves of a knight in a game of chess. The pattern was examined by Kiers, Ten Berge, and Rocci. It will be shown how the knight's move pattern can be found by means of a numeric–algebraic procedure based on the Grobner basis. This approach seems to work almost surely for randomly generated arrays, whether symmetric or nonsymmetric." @default.
• W2019108549 created "2016-06-24" @default.
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• W2019108549 date "2013-11-01" @default.
• W2019108549 modified "2023-09-23" @default.
• W2019108549 title "Direct transformations yielding the knight's move pattern in 3×3×3 arrays" @default.
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• W2019108549 doi "https://doi.org/10.1016/j.chemolab.2013.05.014" @default.
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