Mean-squared-error (MSE) is one of the most widely used performance metrics for the designs and analysis of multi-input-multiple-output (MIMO) communications. Weighted MSE minimization, a more general formulation of MSE minimization, plays an important role in MIMO transceiver optimization. While this topic has a long history and has been extensively studied, existing treatments on the methods in solving the weighted MSE optimization are more or less sporadic and non-systematic. In this paper, we firstly review the two major methodologies, Lagrange multiplier method and majorization theory based method, and their common procedures in solving the weighted MSE minimization. Then some problems and limitations of the methods that were usually neglected or glossed over in existing literature are provided. These problems are fundamental and of critical importance for the corresponding MIMO transceiver optimizations. In addition, a new extended matrix-field weighted MSE model is proposed. Its solutions and applications are discussed in details. Compared with existing models, this new model has wider applications, e.g., nonlinear MIMO transceiver designs and capacity-maximization transceiver designs for general MIMO networks.