Generalized inverse of covariance matrix of conditional distribution of normal random vector

Question

In this article of wiki, it says that if $(X_1',X_2')'$ is (multivariate) normal, the covariance matrix of $X_1|X_2=a$ is $\Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{12}'$, where $\Sigma_{22}^{-1}$ is "the generalized inverse of $\Sigma_{22}$." However, as I understand, the generalized inverse is not unique. Then how can the conditional distribution be thus obtained?