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
$$M=\left(\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \hfill C\hfill & \hfill D\hfill \end{array}\right)$$ 
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, $AB{D}^{1}C$. Analogously if $A$ is invertible then the Schur complement of the block $A$ of the matrix $M$ is the
$q\times q$ matrix, $DC{A}^{1}B$.
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^{},
$$T=\left(\begin{array}{cc}\hfill I\hfill & \hfill O\hfill \\ \hfill {D}^{1}C\hfill & \hfill {D}^{1}\hfill \end{array}\right)$$ 
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
$$T=\left(\begin{array}{cc}\hfill {A}^{1}\hfill & \hfill O\hfill \\ \hfill C{A}^{1}\hfill & \hfill I\hfill \end{array}\right)$$ 
see also:

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

Canonical name  SchurComplement 
Date of creation  20130322 15:27:11 
Last modified on  20130322 15:27:11 
Owner  georgiosl (7242) 
Last modified by  georgiosl (7242) 
Numerical id  8 
Author  georgiosl (7242) 
Entry type  Definition 
Classification  msc 15A15 
Related topic  BlockDeterminants 
Related topic  MatrixInversionLemma 