Day: October 19, 2012

Here are some formula for matrix inversion. Lemma 1: For a block matrix $latex M=\begin{pmatrix}A & B \\C &D\end{pmatrix}$, $latex M^{-1}=\begin{pmatrix}(A-B D^{-1} C)^{-1}& -A^{-1}B(D-CA^{-1}B)^{-1}\\-(D-CA^{-1}B)^{-1}CA^{-1}&(D-CA^{-1}B)^{-1}\end{pmatrix}$ $latex =\begin{pmatrix}A^{-1}+A^{-1}BS^{-1}CA^{-1}& -A^{-1}BS^{-1}\\-S^{-1}CA^{-1}&S^{-1}\end{pmatrix}$, where $latex S=D-CA^{-1}B$ is basically the Schur’s complement of block $latex A$. Proof: Let $latex M^{-1}=\begin{pmatrix}E&F\\G&H\end{pmatrix}$, $latex M M^{-1}=1$ gives us $latex AE+BG=I$ $latex AF+BH=0$ $latex CE+DG=0$ $latex…