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The Gray and Wyner lossy source coding for a simple network for sources thatgenerate a tuple of jointly Gaussian random variables (RVs) $X_1 : \Omega\rightarrow {\mathbb R}^{p_1}$ and $X_2 : \Omega \rightarrow {\mathbbR}^{p_2}$, with respect to square-error distortion at the two decoders isre-examined using (1) Hotelling's geometric approach of Gaussian RVs-thecanonical variable form, and (2) van Putten's and van Schuppen'sparametrization of joint distributions ${\bf P}_{X_1, X_2, W}$ by Gaussian RVs$W : \Omega \rightarrow {\mathbb R}^n $ which make $(X_1,X_2)$ conditionallyindependent, and the weak stochastic realization of $(X_1, X_2)$. Item (2) isused to parametrize the lossy rate region of the Gray and Wyner source codingproblem for joint decoding with mean-square error distortions ${\bfE}\big\{||X_i-\hat{X}_i||_{{\mathbb R}^{p_i}}^2 \big\}\leq \Delta_i \in[0,\infty], i=1,2$, by the covariance matrix of RV $W$. From this then followsWyner's common information $C_W(X_1,X_2)$ (information definition) is achievedby $W$ with identity covariance matrix, while a formula for Wyner's lossycommon information (operational definition) is derived, given by$C_{WL}(X_1,X_2)=C_W(X_1,X_2)  = \frac{1}{2} \sum_{j=1}^n  \ln  \left(  \frac{1+d_j}{1-d_j}  \right),$ for the distortion region $ 0\leq \Delta_1 \leq\sum_{j=1}^n(1-d_j)$, $0\leq \Delta_2 \leq \sum_{j=1}^n(1-d_j)$, and where $1 >d_1 \geq d_2 \geq \ldots \geq d_n>0$ in $(0,1)$ are {\em the canonicalcorrelation coefficients} computed from the canonical variable form of thetuple $(X_1, X_2)$. The methods are of fundamental importance to other problemsof multi-user communication, where conditional independence is imposed as aconstraint.

Characterization of Conditional Independence and Weak Realizations of Multivariate Gaussian Random Variables: Applications to Networks