Bayesian updating of multivariate normal?

Question

Let $\bf x$ be an unobserved realization of $\tilde{\bf x}\sim\mathcal{N}(\pmb\mu,\pmb\Sigma)$, where $\pmb\mu\equiv\begin{bmatrix}\mu_1\\\mu_2\end{bmatrix}$ and $\pmb\Sigma\equiv\begin{bmatrix}\sigma_1^2&\sigma_{12}\\\sigma_{12}&\sigma_2^2\end{bmatrix}$. We observe a signal $\bf y=\bf x+\bf u$, where $\bf u$ is an unobserved realization of $\tilde{\bf u}\sim\mathcal{N}(\pmb\varepsilon,\pmb\Psi)$, where $\pmb\varepsilon\equiv\begin{bmatrix}0\\0\end{bmatrix}$ and $\pmb\Psi\equiv\begin{bmatrix}\psi^2_1&0\\0&\psi^2_2\end{bmatrix}$. What is the distribution of $\bf x$ conditional on $\{\pmb\mu,\pmb\Sigma,\bf y,\pmb\varepsilon,\pmb\Psi\}$?