Let S =  X^TX be the sample covariance matrix where L and P are latent roots and vectors of X^TX respectively. Prove that S has latent root  and latent vectors P. Using this result show that correlation loadings of S are given by

where s_i is the ith diagonal element of S, and consequently correlation loadings are the same for both X^TX and S. Show also that X^TX and S possess the same standardized PC scores, but that unstandardized scores differ.