# Schur complement

Let A,B,C,D be matrices of sizes $p\times p$, $p\times q$, $q\times p$ and $q\times q$ respectively and suppose that $D$ is invertible. Let

 $M=\begin{pmatrix}A&B\\ C&D\end{pmatrix}$

so that $M$ is a $(p+q)\times(p+q)$ matrix.
Then the Schur complement of the block $D$ of the matrix $M$ is the $p\times p$ matrix, $A-BD^{-1}C$. Analogously if $A$ is invertible then the Schur complement of the block $A$ of the matrix $M$ is the $q\times q$ matrix, $D-CA^{-1}B$. In the first case, when $D$ is invertible, the Schur complement arises as the result of performing a partial Gaussian elimination by multiplying the matrix $M$ from the right with the lower triangular block matrix,

 $T=\begin{pmatrix}I&O\\ -D^{-1}C&D^{-1}\end{pmatrix}$

where $I$ is the $p\times p$ identity matrix and $O$ is the $p\times q$ zero matrix. Analogously, in the second case, we take the Schur complement by multiplying the matrix $M$ from the left with the lower triangular block matrix

 $T=\begin{pmatrix}A^{-1}&O\\ -CA^{-1}&I\end{pmatrix}$