  The capacity and optimal signaling over a fixed Gaussian MIMO channel are
considered under the joint total and per-antenna power constraints (TPC and
PAC). While the general case remains an open problem, a closed-form full-rank
solution is obtained along with its sufficient and necessary conditions. The
conditions for each constraint to be inactive are established. The high and
low-SNR regimes are studied. Isotropic signaling is shown to be optimal in the
former case while rank-1 signaling (beamforming) is not necessarily optimal in
the latter case. Unusual properties of optimal covariance under the joint
constraints are pointed out.
