= Schur complement
{c}
{title2=$S=D-CA^{-1}B$}
{wiki}
For a <block matrix> $M=\begin{pmatrix}A&B\\C&D\end{pmatrix}$ with $A$ an <invertible matrix>, the <Schur complement> of $A$ in $M$ is $S=D-CA^{-1}B$. It is the effective coefficient <matrix> for the remaining variables after <block elimination>. The definition requires invertibility of $A$, but not of $S$ or $M$. Its size is that of $D$, even when $B$ and $C$ are rectangular <matrices>.
In the original example under <block matrix inverse>, $D=5$ becomes $S=19/6$ because changing the remaining variable also changes the two variables solved through $A$. This is what makes a <Schur complement> more informative than simply deleting rows and columns. It retains their mediated effect, and its <matrix inverse> is the corresponding block of $M^{-1}$ when the <block matrix inverse> exists.
The same elimination appears when minimizing a <quadratic form>. The <Schur complement as a minimized quadratic form> gives a geometric example; the <Schur complement determinant identity> gives an exact test for an <invertible matrix>.
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