I have been reading the textbook Elements of Information Theory and was hoping to get some clarification about a small section of the textbook. When deriving the rate-distortion function for independent Gaussian variables with different variances, the author uses a reverse water-filling technique. Very interesting technique, however, I am interested in deriving the rate distortion for variables i.i.d from a multivariate Gaussian distribution i.e. $X_1, ..., X_m \sim N(0, \Sigma) : \Sigma \ $ is a n x n matrix. The authors say:
More generally, the rate distortion function for a multivariate normal vector can be obtained by reverse water-filling on the eigenvalues.
I am unsure what they exactly mean by this. Water-filling on what eigenvalues? I'm assuming they mean that of the covariance matrix but it seems likely that it could also mean the eigenvalues of the sample covariance matrix. If anyone can help elucidate the exact procedure for calculating this rate distortion function for a multivariate Gaussian, I would be forever indebted to you.