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

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

see also:

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

Title	Schur complement
Canonical name	SchurComplement
Date of creation	2013-03-22 15:27:11
Last modified on	2013-03-22 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