Suppose $(X, Y)$ is a zero mean Gaussian vector with $X$ and $Y$ both vectors in their own right. Define $\Sigma_Y = E(YY')$, $\Sigma_{XY} = E(XY')$. In the case where $\Sigma_Y$ is invertible, we easily see that $$(X - \Sigma_{XY} \Sigma_Y^{-1}Y, Y)$$ is also Gaussian (as a linear transformation of $(X,Y)$) with zero covariance between its two parts (so that they are independent) and thus $$E(X |Y) = \Sigma_{XY} \Sigma_Y^{-1}Y$$
I am very confused however, as to what happens when $\Sigma_Y$ is NOT invertible. Is $E(X |Y)$ still a linear transformation of $Y$? If so, what would it be? Does this somehow involve the Moore-Penrose inverse of $\Sigma_Y$?