 Solution

Write the partitioned mean as $({\mu }_1^T,{\mu }_2^T{)}^T$ and assume $V_{22}$ is nonsingular, as required by the displayed inverse. For the conditional multivariate normal distribution, form the residualA linear transformation of a multivariate normal distribution is again normal. Direct calculation givesThe jointly normal blocks $U$ and $X_2$ are therefore independent: their joint characteristic function has no cross term in its quadratic exponent and factors into the two marginal characteristic functions. Conditioning leaves the distribution of $U$ unchanged. Reconstructing $X_1$ proves the full conditional law, including the requested Schur complement covariance:This describes a regular conditional law; it does not require the equality event to have positive probability.