Schur complement ($S=D-C{A}^{-1}B$)

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.

Let $M=\left(\begin{array}{cc}A& B\\ B^T& D\end{array}\right)$ be a real symmetric matrix with $A$ a positive-definite matrix. Its quadratic form satisfies the completing the square identityFor each fixed vector $y$, the unique minimizing vector is $x=-A^{-1}By$, so the minimized value is $y^{T}Sy$. Also, $M$ is a positive-definite matrix exactly when $S$ is a positive-definite matrix, under the stated assumption on $A$.^[ref]

An original scalar example isHere $A=2$, $B=1$, $D=3$, and the Schur complement is $5/2$. Holding $x=0$ would give $3y^2$, but letting $x$ relax to $-y/2$ reduces the quadratic form to $(5/2)y^2$. At $y=2$ the respective values are $12$ and $10$. Eliminating a variable therefore changes the effective cost of the retained variable.

This interpretation supports quadratic optimization and the positive-definite symmetric matrix tests used in Cholesky decomposition. In a multivariate normal distribution, a covariance matrix also gives a conditional covariance through a Schur complement; the required probabilistic hypotheses are supplied by conditional multivariate normal distribution, rather than by the algebra alone.

When $A$ is an invertible matrix, the factorization in block elimination has outer triangular factors with determinant one. Hence $\det M=\det A\det S$, without a symmetry assumption. For the original example under block matrix inverse, this gives $\det M=(2\cdot 3)(19/6)=19$. An invertible block $A$ therefore reduces the question of whether the full matrix is an invertible matrix to the smaller Schur complement; a singular $S$ makes $M$ a singular matrix.

For a Hermitian matrix, let $x_i$ be its $i$th column without the diagonal entry and let $G^{(i)}=(X^{(i)}-zI{)}^{-1}$. Solve the two block equations of the inverse against the $i$th coordinate vector to obtain this formula. Both full and minor inverses must exist; nonreal $z$ guarantees that. In the real symmetric case $x_i^{\ast }$ may be replaced by $x_i^T$.