We consider a distributed source coding problem of $L$ correlated Gaussian
observations $Y_i, i=1,2,...,L$. We assume that the random vector
$Y^{L}={}^{\rm t} (Y_1,Y_2,$ $...,Y_L)$ is an observation of the Gaussian
random vector $X^K={}^{\rm t}(X_1,X_2,...,X_K)$, having the form $Y^L=AX^K+N^L
,$ where $A$ is a $L\times K$ matrix and $N^L={}^{\rm t}(N_1,N_2,...,N_L)$ is a
vector of $L$ independent Gaussian random variables also independent of $X^K$.
The estimation error on $X^K$ is measured by the distortion covariance matrix.
The rate distortion region is defined by a set of all rate vectors for which
the estimation error is upper bounded by an arbitrary prescribed covariance
matrix in the meaning of positive semi definite. In this paper we derive
explicit outer and inner bounds of the rate distortion region. This result
provides a useful tool to study the direct and indirect source coding problems
on this Gaussian distributed source coding system, which remain open in
general.