  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.
