The conventional Gaussian multiple-input multipleoutput (MIMO) broadcast channel (BC)- multiple-access channel (MAC) duality has previously been applied to solve non-convex BC capacity computation problems. However, this conventional duality approach is applicable only to the case in which the base station (BS) of the BC is subject to a single sum-power constraint. An alternative approach is the minimax duality, established by Yu in the framework of Lagrange duality, which can be applied to solve the per-antenna power constraint case. This paper first extends the conventional BC-MAC duality to the general linear transmit covariance constraint (LTCC) case, and thereby establishes a general BC-MAC duality. This new duality is then applied to solve the BC capacity computation problem with multiple LTCCs. Moreover, the relationship between this new general BC-MAC duality and the minimax duality is also presented, and it is shown that the general BC-MAC duality has a simpler form. Numerical results are provided to illustrate the effectiveness of the proposed algorithm.