  In this letter, we investigate an important and famous issue, namely weighted
mean-square-error (MSE) minimization transceiver designs. In our work, for
transceiver designs a novel weighted MSE model is proposed, which is defined as
a linear matrix function with respect to the traditional data detection MSE
matrix. The new model can be interpreted an extension of weighting operation
from vector field to matrix field. Based on the proposed weighting operation a
general transceiver design is proposed, which aims at minimizing an increasing
matrix-monotone function of the output of the previous linear matrix function.
The structure of the optimal solutions is also derived. Furthermore, two
important special cases of the matrix-monotone functions are discussed in
detail. It is also revealed that these two problems are exactly equivalent to
the transceiver designs of sum MSE minimization and capacity maximization for
dual-hop amplify-and-forward (AF) MIMO relaying systems, respectively. Finally,
it is concluded that the AF relaying is undoubtedly this kind of weighting
operation.
