Distortion-rate tradeoff of a source uniformly distributed over positive semi-definite matrices

We study the distortion-rate tradeoff D(K) of a source uniformly distributed over a manifold comprising of positive semi-definite matrices - over both real and complex fields - under a trace constraint. This models the impact exercised by limited-rate feedback on the information transfer of optimal dasiainput covariance matricespsila to a transmitter from a channel-aware receiver in a multiple-input mulitple-output (MIMO) system. Utilizing the volume of a geodesic ball in the manifold, D(K) is bounded between asymptotically tight lower and upper limits obtained via sphere-packing and random coding arguments, respectively.