While trying to simplify some expressions involving Gaussian distributions I have come to the following. The block matrix $K = \begin{bmatrix} K_{11} & K_{12} \\ K_{21} & K_{22} \end{bmatrix}$ is a covariance matrix, so is symmetric and positive semi-definite. Let $\Lambda = K^{-1}$ and break it up into blocks in the same way as $K$, i.e. $\Lambda = \begin{bmatrix} \Lambda _{11} & \Lambda _{12} \\ \Lambda_{21} & \Lambda _{22} \end{bmatrix}$.
I have the expression $(\Lambda_{22} - \Lambda_{21}(\Lambda_{11} + \sigma^{-2}I)^{-1}\Lambda_{12})^{-1}$ where I expect to obtain $K_{22} - K_{21}(K_{11} + \sigma^{2}I)^{-1}K_{12}$ - this comes in a derivation of a conditional distribution.
These two expressions seem to actually be equivalent, after checking a few random examples, but I've not been able to show this. I see that the first expression is the Schur complement of $\Lambda_{11} + \sigma^{-2}I$ in $\begin{bmatrix} \Lambda _{11} + \sigma^{-2}I & \Lambda _{12} \\ \Lambda_{21} & \Lambda _{22} \end{bmatrix}$, and the latter is the Schur complement of $K_{11} + \sigma^2I $ in $\begin{bmatrix} K_{11} + \sigma^2I & K_{12} \\ K_{21} & K_{22} \end{bmatrix}$, but I cannot see a good reason for these to be equivalent.