Retrieving the correlation matrix from a truncated PCA solution: The inverse principal component problem

When r Principal Components are available for k variables, the correlation matrix is approximated in the least squares sense by the loading matrix tim es its transpose. The approximation is generally not perfect unless r = k. In the present paper it is shown that, when r is at or above the Ledermann bound, r principal components are enough to perfectly reconstruct the corre lation matrix, albeit in a way more involved than taking the loading matrix times its transpose. In certain cases just below the Ledermann bound, reco very of the correlation matrix is still possible when the set of all eigenv alues of the correlation matrix is available as additional information.