This paper considers the optimal transmit strategy for multi-antenna bidirectional broadcast channels with per-antenna power constraints. First, an equivalent formulation of the weighted rate sum maximization problem is provided. This allows us to come up with an effective solution to characterize the boundary of the capacity region which relies on the weighted rate sum optimal rate pair. To that end, an iterative algorithm to find the optimal transmit strategy is derived, the convergence to the optimum is proved, and a closed-form solution of the corresponding off-diagonal elements of the optimal transmit strategy is provided. Further, we provide a parametrization of the curved section of the capacity region. Finally, the theoretical results and algorithm performance are illustrated by numerical examples.