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

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, $A-B{D}^{-1}C$. Analogously if $A$ is invertible then the Schur complement of the block $A$ of the matrix $M$ is the$q\times q$ matrix, $D-C{A}^{-1}B$.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,

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

see also:

• Wikipedia, http://en.wikipedia.org/wiki/Schur_complementSchur complement