For multivariate normal X∼N(μ,Σ)\mathbf{X} \sim N(\boldsymbol{\mu}, \boldsymbol{\Sigma})X∼N(μ,Σ), the conditional covariance Cov(X1∣X2)Cov(\mathbf{X}_1|\mathbf{X}_2)Cov(X1∣X2) equals:
Σ11−Σ12Σ22−1Σ21\boldsymbol{\Sigma}_{11} - \boldsymbol{\Sigma}_{12}\boldsymbol{\Sigma}_{22}^{-1}\boldsymbol{\Sigma}_{21}Σ11−Σ12Σ22−1Σ21
Σ11\boldsymbol{\Sigma}_{11}Σ11
Σ22\boldsymbol{\Sigma}_{22}Σ22
Σ12Σ22−1\boldsymbol{\Sigma}_{12}\boldsymbol{\Sigma}_{22}^{-1}Σ12Σ22−1