Given the left hand side of the equation below, how to show the equality with the right hand side? I checked it numerically, but not sure how to prove it.
\begin{align} A - AC^T(CAC^T +R)^{-1} CA = (A^{-1}+ C^T R^{-1} C)^{-1} \end{align}
$A$ is an invertable square matrix.
$C$ is a rectangular matrix with compatible dimension w.r.t $A$ and $R$
$R$ is an invertable ssquare matrix.