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1325 days ago Leave a comment Permalink

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}

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