This paper studies optimal transmit strategies for multiple-input multiple-output (MIMO) Gaussian channels with joint sum and per-antenna power constraints. It is shown that if an unconstraint optimal allocation for an antenna exceeds a per-antenna power constraint, then the maximal power for this antenna is used in the constraint optimal transmit strategy. This observation is then used in an iterative algorithm to compute the optimal transmit strategy in closed-form. Finally, a numerical example is provided to illustrate the theoretical results.