I want to know the relationship between the eigensystems of two non-negative-definite (covariance-like) matrices. Both are derived from X which is a T-by-K real matrix (wlog say K > T). I avoid subtracting the mean to avoid complicating the main issue which is about the eigensystem structure.
Ct = (1/T) * X' X = Et Lt Et'
Ck = (1/K) * X X' = Ek Lk Ek'
.... where Et, Lt are the (column) eigenvectors and diagonal matrix of eigenvalues respectively for Ct, and Ek Lk are the same for Ck.
I remember vaguely from school that there is some kind of duality relationship here (between the two eigensystems) which is particularly interesting because both matrices have rank T < K, and hence only T of the diagonal elements of Lt are nonzero.
Does anyone know this relationship explicitly? Thanks in advance for any light you can shed!