Could principle component decomposed coordinates value be correlated to each other? I am wondering if we have a A= n*p matrix of samples and we run a PC decomposition on it. 

Say the eigenvector matrix is E, so the samples in the eigenvector space should be 


  B= A*(E)^-1


So, I am wondering if there is any rule that the columns of B to be correlated? That is, the first PC loadings and the second PC loadings to be correlated?
 Thanks!
PCs are calculated based on eigenvectors (e1 nad e2 in your case) of correlation or covariance matrix. Eigenvectors are orthogonal, so PCs are uncorrelated. If you would like to check the proof, you can find it on page 432 in Applied Multivariate Statistical Analysis 6th edition written by Richard A. Johnson and Dean W. Wichern.