Typical rank and indscal dimensionality for symmetric three-way arrays of order Ix2x2 or Ix3x3

A peculiar property of three-way arrays is that the rank they typically have does not necessarily coincide with the maximum possible rank, given their order. Typical tensorial rank has much been studied over algebraically closed fields. However, very few results have been found pertaining to the typical rank of three-way arrays over the real field. These results refer to arrays sampled randomly from continuous distributions. Arrays that consist of symmetric slices do not fit into this sampling scheme. The present paper offers typical rank results (over the real field) for arrays, containing symmetric slices of order 2 x 2 and 3 x 3. Symmetric arrays often appear to have lower typical ranks than their asymmetric counterparts. This paper also examines whether or not the rank of a symmetric array coincides with the smallest number of dimensions that allow a perfect fit of INDSCAL. For all cases considered, this is indeed true. Thus, a full INDSCAL solution may require fewer dimensions for a symmetric array than a full CP decomposition applied to an asymmetric array of the same size. The reverse situation does not seem to arise. Next, we examine in which cases CP solutions inevitably are INDSCAL solutions. Finally, the rank-reducing impact of double standardizing the slices is discussed. (C) 2004 Elsevier Inc. All rights reserved.