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In this work the transmit covariance matrix optimization problem for the discrete memoryless MIMO Gaussian bidirectional broadcast channel is studied. A half-duplex relay node establishes bidirectional communication between two nodes using a decode-and-forward protocol. In the initial multiple access phase both nodes transmit their messages to the relay node. In the succeeding phase the relay broadcasts an optimal re-encoded message so that both nodes can decode the other's message using their own message as side information. The capacity region of the bidirectional broadcast channel is completely characterized by a weighted rate sum maximization problem, which can be solved by a simple iterative fixed point algorithm. If an efficient transmit covariance matrix is invariant with respect to the joint subspace spanned by the channels, then different combinations of the part transmitted on the orthogonal subspaces result in equivalent transmit strategies with different ranks. A closed-form procedure to obtain the optimal transmit covariance is derived for the case where the rank of the channels is equal to the number of antennas at the relay node and a full-rank transmission is optimal. It shows the complicated structure of the optimal eigenspace, which depends on the weights and the mean transmit power constraint. For parallel channels the optimal solution is completely characterized and discussed, which also solves the optimal power allocation problem for a single-antenna OFDM system.