When there exists the limitation of communication bandwidth between sensors and a fusion center, one needs to optimally pre-compress sensor outputs-sensor observations or estimates before sensors' transmission in order to obtain a constrained optimal estimation at the fusion center in terms of the linear minimum error variance criterion; or when an allowed performance loss constraint exists, one needs to design the minimum dimension of sensor data. This paper will answer the above questions by using the matrix decomposition, pseudo-inverse, and eigenvalue techniques.