given $\begin{bmatrix}X\\Y\end{bmatrix}\sim N\left(\begin{bmatrix}\mu_X\\\mu_Y\end{bmatrix};\begin{bmatrix}\Sigma_{XX}&\Sigma_{XY}\\\Sigma_{YX}&\Sigma_{YY}\end{bmatrix}\right)$ How can I prove that $\Sigma_{YX}-\Sigma_{YX}\Sigma_{XX}^{-}\Sigma_{XX}=0$, where $\Sigma_{XX}^{-}$ is the reflexive g-inverse, i.e. $\Sigma_{XX}\Sigma_{XX}^{-}\Sigma_{XX}=\Sigma_{XX}$.

The bigger picture is that I am trying to find en expression for the conditional distribution $Y|X=x$ when $\Sigma_{XX}$ is singular. My approach is to prove that $X$ and $Y-\Sigma_{YX}\Sigma_{XX}^{-}X$ are independent. Once this is proved the rest is trivial. However, I got stuck with the problem above.