When there exists the limitation of communication bandwidth between sensors and a fusion center, one needs to optimally pre-compresssensor outputs-sensor observations or estimates before sensors'transmission to obtain a constrained optimal estimation at the fusion center interms of the linear minimum error variance criterion. This paper will give an analytic solution of the optimal linear dimensionality compressionmatrix for the single sensor case and analyze the existence of the optimal linear dimensionality compression matrix for the multisensor case, as well as how to implement a Gauss-Seidel algorithm to search for an optimal solution to linear dimensionality compression matrix.