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Write a linear system as $Ax+By=f$ and $Cx+Dy=g$, with $A$ an invertible matrix. Solving the first equation for $x$ and substituting in the second givesThus the Schur complement is the coefficient matrix remaining after the $x$ variables have been eliminated. When it is an invertible matrix, solve the reduced linear system for $y$ and recover $x$. In computation, expressions such as $A^{-1}B$ mean solving with each column of $B$, using the same LU decomposition or Cholesky decomposition when applicable.

For the example under block matrix inverse and right-hand side $(3,1,4{)}^T$, take $f=(3,1{)}^T$ and $g=4$. Since $C{A}^{-1}f=13/6$, the reduced equation is $(19/6)y=11/6$. Back substitution givesThe original equations return $3$, $1$ and $4$, respectively. The reduction retained the influence of $x_1,x_2$ on $y$ through the correction $C{A}^{-1}B$.

A short proof outline also explains whether $M$ is an invertible matrix: the exact matrix factorizationhas invertible matrices on the outside. Therefore $M$ is an invertible matrix precisely when the central block diagonal matrix is one. Reversing the factors gives the block matrix inverse, while taking determinants gives the Schur complement determinant identity.

For a block matrix $M=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$ with $A$ an invertible matrix, the Schur complement of $A$ in $M$ is $S=D-C{A}^{-1}B$. 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.