There are numerous criteria to choose how many Principal Components should be retained when reducing dimensionality via a Principal Component Analysis. Subjective decisions by the user, and/or distributional hypotheses are usually necessary when applying these criteria. This communication proposes a new criterion which avoids these problems. The criterion is based on geometric properties of the cone of positive semi-definite matrices, that suggest a smallest admissible dimensionality. The structure of the cone, which is associated with the use of the usual matrix inner product, precludes the presence of lowrank matrices in certain areas of the cone.

keywords: principal components, factorial analysis, dimensionality reduction, cone of positive semidefinite matrices.