# How to calculate the variance-covariance matrix of the principal components from the variance-covariance matrix of the original data?

This is a PCA question. I am reading about the mathematics behind PCA on this website. I understand that Y = XA is the matrix notation of the transformation of the original variables to the principal components where X is the feature vector and rows of the A matrix represent the eigenvectors and within each row we have the loading.

Then using the A matrix and the Sx matrix (the var-covar matrix of the original data), we can derive the var-covar matrix of the PC.

i.e.

I am not sure what linear algebra method this is called and it derives the var-covar matrix of the PCs, which is called Sy and the elements in the diagonal of this matrix are the eigenvalues and that is the variance explained by each principal component. If that is the case, we will expect the very first element of the matrix to be the variance of the first component which should be the largest.

If this is not the way, then how do we calculate the variance-covariance matrix of the principal components and what does this matrix tell us?