For any multiple-input multiple-output (MIMO) network with linear beamformers, it is shown that there exists a dual network that attains the same signal-to-interference-plus-noise ratio (SINR) performance. This network duality concept is a generalization of the virtual uplink concept investigated by Rashid-Farrokhi . We first develop the dual relation using a generic duality theory in linear programming. This approach naturally leads to the construction of a dual network, and provides machinery for handling a generalized cost function. We then consider the joint MIMO beamforming and power control problem with individual SINR constraints. We apply the network duality to this problem and propose a high-performance iterative algorithm which, compared with past centralized approaches, has improved convergence behavior. Finally, we propose a simpler distributed version of the algorithm tailored for a cellular downlink. This algorithm can be readily implemented in a distributed fashion, since it does not require information exchange between base stations. The algorithm achieves performance close to that of centralized ones, and outperforms other decentralized approaches available in the literature