Symmetry transformations for square sliced three-way arrays, with applications to their typical rank
The typical tensorial rank of three-way arrays over algebraically closed fields has been much studied. In this paper, the tensorial rank over the real field is examined. The possible values of typical rank of some \(p \times q \times r\) arrays are given. Especially, the typical rank of arrays with square slices, i.~e., with \(q = r\) are analyzed. When \(p = 2\), square slice arrays have typical rank \([q, q+1]\). Special attention is given to square cases with \(p > 2\). The bounds for the typical rank of such arrays are derived and it is proved that the typical rank of the ``cubic'' \(3 \times 3 \times 3\) array is \(5\). The main result also allows, for \(p \times q \times r\) arrays with \(p > 2\), to discard \(q + 1\) as a value of positive probability, except when \(p = 3\) and \(q = 2\). This is done by introducing a symmetry criterion.
scientific article; zbMATH DE number 5072863