 Solution

For disjoint nonempty index sets $A$ and $B$, partition the mean and covariance conformably. The conditional multivariate normal distribution isIf the sets overlap, the shared coordinates are fixed and this formula applies to $A\setminus B$.

Partition ${\Omega }_0$ into the $A$ and $A^c$ blocks. The block-inverse formula identifies the displayed conditional covariance withFor $A=\{j,k\}$, inversion of the two-by-two precision block givesThe denominator is positive, so the conditional covariance vanishes exactly when ${\Omega }_{0,jk}=0$. A Gaussian pair is independent exactly when it is uncorrelated, proving

Profiling the Gaussian likelihood over $\mu$ gives $\hat{\mu }=\bar{X}$. Apart from constants and a positive factor, the negative log-likelihood for the precision matrix isIts derivative is $-{\Omega }^{-1}+S$, so the unpenalized minimizer is $S^{-1}$. The Graphical Lasso solveswith some conventions also penalizing the diagonal.