## Block Matrix Inverse

Previously I wrote a post on the Woodbury matrix inverse formula. One of the results derived was $\begin{pmatrix} V_{11} & V_{12}\\ V_{21} & V_{22}\\ \end{pmatrix}^{-1} = \begin{pmatrix} V_{11.2}^{-1} & -V_{11.2}^{-1}V_{12}V_{22}^{-1}\\ -V_{22}^{-1}V_{21}V_{11.2}^{-1} & V_{22}^{-1}+V_{22}^{-1}V_{21}V_{11.2}^{-1}V_{12}V_{22}^{-1} \end{pmatrix}$ $\begin{pmatrix} V_{11} & V_{12}\\ V_{21} & V_{22}\\ \end{pmatrix}^{-1} =\begin{pmatrix} V_{11}^{-1}+V_{11}^{-1}V_{12}V_{22.1}^{-1}V_{21}V_{11}^{-1} & -V_{11}^{-1}V_{12}V_{22.1}^{-1}\\ -V_{22.1}^{-1}V_{21}V_{11}^{-1} & V_{22.1}^{-1} \end{pmatrix}$

Hence the inverse of a block matrix can be written neatly in one of two ways: $\begin{pmatrix} V_{11} & V_{12}\\ V_{21} & V_{22}\\ \end{pmatrix}^{-1} = \begin{pmatrix} V_{11.2}^{-1} & -V_{11.2}^{-1}V_{12}V_{22}^{-1}\\ -V_{22}^{-1}V_{21}V_{11.2}^{-1} & V_{22.1}^{-1} \end{pmatrix}$ $\begin{pmatrix} V_{11} & V_{12}\\ V_{21} & V_{22}\\ \end{pmatrix}^{-1} =\begin{pmatrix} V_{11.2}^{-1} & -V_{11}^{-1}V_{12}V_{22.1}^{-1}\\ -V_{22.1}^{-1}V_{21}V_{11}^{-1} & V_{22.1}^{-1} \end{pmatrix}$