Consider the following Multivariate Normal distribution:
\[\begin{aligned} \begin{pmatrix} \bm u \\ \bm v \end{pmatrix} &\sim \MvNormal\p{ \begin{bmatrix} \bm \mu_u \\ \bm \mu_v \end{bmatrix}, \begin{bmatrix} \bm \Sigma_u & \bm \Sigma_{uv}\\ \bm \Sigma_{vu} & \bm \Sigma_v \end{bmatrix} }. \end{aligned}\]Then,
\[\bm u \mid \bm v \sim \MvNormal( \bm\mu_u + \bm\Sigma_{uv} \bm\Sigma_v ^{-1} (\bm v - \bm\mu_v), \bm\Sigma_u - \bm\Sigma_{uv} \bm\Sigma_v^{-1} \bm\Sigma_{vu} )\]