Linear contraction of a two-coordinate Gaussian Gibbs sweep ($b_{k+1}-{\mu }_b=\frac{C^2}{AD}(b_k-{\mu }_b)+{\eta }_k$)

For Gaussian precision $\left(\begin{array}{cc}A& C\\ C& D\end{array}\right)$, a systematic update of the first coordinate followed by the second makes the centered second coordinate an AR(1) chain with coefficient $C^2/(AD)<1$. Independent conditional Gaussian noise supplies its stable innovation. This proves convergence of the Gaussian Gibbs sampler and explains slower mixing when posterior correlation has magnitude near one.