In this paper,we consider the problem of minimizing sum power under the constraint of a sum rate in fading Gaussian multiple input multiple output(MIMO) broadcast channel(BC) with additive Gaussian noise.This problem arises frequently in dynamic resource allocation of wireless communication systems.It is the complementary problem to sum rate maximization with a sum power constraint for fading MIMO downlink.We first formulate the equivalent convex optimization problem using the duality between the MIMO multi-access channel(MAC) and the MIMO BC.Then we derive a simple and fast iterative water-filling algorithms based on the subgradient and bisection methods that compute sum power of the transmitter.Theory analysis and numerical simulations show that the proposed algorithm converges to the minimum sum power globally and efficiently.