I know the generally reasons of using correlation matrix vs a covariance matrix when doing PCA (and visa versa) however when thinking about the eigenvectors (principal components of the data) of each of these matrices I still have a question which is:
Obviously, these eigenvectors will be different however since we are just scaling one to get the other is there just a 1:1 mapping to the eigenvectors of a correlation matrix to a covariance matrix? I.e are the principal components we get for one conceptually the same just in different coordinate system or are there total different interpretations for each set of principal components we get when doing PCA with correlation or covariance?