# Inverse of $I+MW$

According Constrained clustering via affinity propagation, the inverse of $I+MW$ can be computed in closed form using $$\begin{bmatrix}A & B \\ 0 & I\end{bmatrix}^{-1} = \begin{bmatrix}A^{-1} & -A^{-1}B \\ 0 & I\end{bmatrix}$$ where $M$ and $W$ are symmetric matrices and $A$ is apparently a $2L\times 2L$ matrix where $L$ is the number of non-zeros entries in the upper triangle of $M$ (discounting the diagonal).

Anyone can explain me how this works ?

• It could be referring to the particular $M$ and $W$ in the paper ($K$ and $M$). – Kasra Dec 19 '16 at 14:10

You may try $$\begin{pmatrix} A & B \\ 0 & I \end{pmatrix}\begin{pmatrix} W & X \\ Y & Z \end{pmatrix}=\begin{pmatrix} I & 0 \\ 0 & I \end{pmatrix}$$Then expand left hand side and compare terms, find W, X, Y, Z.