A (Very) Useful Formula for Matrix Inversion

Here is a very useful formula for matrix inverse, I think the first time I saw it is from Prof. John Cioffi's book.

\[(A+BD^{-1}C)^{-1}=A^{-1}-A^{-1}B(D+CA^{-1}B)^{-1}CA^{-1}\]

let me give one example of how it can be used. Assuming that \(R_{uu}=h_{2}h_{2}^{h}+\lambda I\) where \(h_{2}\) is a vector. Then we have

\[R_{uu}^{-1}=\lambda^{-1}I-\frac{1}{\lambda^{2}+\lambda h_{2}^{h}h_{2}}h_{2}h_{2}^{h}\]

By using this formula, the end result turns out to be much simpler than where it starts from.