This paper studies linear transmit filter design for weighted sum-rate (WSR) maximization in the multiple input multiple output broadcast channel (MIMO-BC). The problem of finding the optimal transmit filter is non-convex and intractable to solve using low complexity methods. Motivated by recent results highlighting the relationship between mutual information and Minimum Mean Square Error (MMSE), this paper establishes a relationship between weighted sum-rate and weighted MMSE in the MIMO-BC. The relationship is used to propose a low complexity algorithm for finding a local weighted sum-rate optimum based on alternating optimization. Numerical results studying sum-rate show that the proposed algorithm achieves high performance with few iterations.