Schur complement
The Schur complement is a matrix expression obtained by eliminating one diagonal block of a block matrix. For a matrix
[ M= \begin{pmatrix} A & B\ C & D \end{pmatrix}, ]
where (A) and (D) are square matrices, the Schur complement of (D) in (M), provided that (D) is invertible, is
[ M/D=A-BD^{-1}C. ]
If (A) is invertible, the Schur complement of (A) in (M) is correspondingly
[ M/A=D-CA^{-1}B. ]
The construction expresses the effect of one group of variables after another group has been algebraically eliminated. It therefore connects Gaussian elimination, block factorization, determinant identities, quadratic forms, conditional probability distributions, and constrained optimization.
Historical development
The determinant reduction underlying the Schur complement appeared in nineteenth-century work on linear equations and partitioned determinants. In 1917, Issai Schur and You Watanabe formulated the block reduction in their analysis of determinant inequalities and representations of matrices, giving the expression its modern algebraic setting.
The term “Schur complement” was introduced by Emilie Virginia Haynsworth during the mid-twentieth-century development of partitioned-matrix theory. Haynsworth also established systematic connections between Schur complements, matrix inertia, and definiteness. These results transformed an elimination formula into a standard object of matrix analysis.
Block elimination
The defining expression follows directly from block Gaussian elimination. When (D) is invertible, the matrix (M) admits the factorization
[ \begin{pmatrix} A & B\ C & D \end{pmatrix}
\begin{pmatrix} I & BD^{-1}\ 0 & I \end{pmatrix} \begin{pmatrix} A-BD^{-1}C & 0\ C & D \end{pmatrix}. ]
An equivalent symmetric arrangement is
[ M= \begin{pmatrix} I & 0\ CD^{-1} & I \end{pmatrix} \begin{pmatrix} A-BD^{-1}C & B\ 0 & D \end{pmatrix}. ]
The triangular factor has determinant one, so the determinant of (M) separates as
[ \det M=\det D\det(A-BD^{-1}C). ]
When (A) is invertible, elimination in the opposite order gives
[ \det M=\det A\det(D-CA^{-1}B). ]
These identities generalize the determinant formula for a scalar (2\times2) matrix. They also show that the Schur complement is not merely a selected submatrix: it includes the correction produced by coupling through the eliminated block.
For the block system
[ \begin{pmatrix} A & B\ C & D \end{pmatrix} \begin{pmatrix} x\ y \end{pmatrix}
\begin{pmatrix} f\ g \end{pmatrix}, ]
the second block equation yields (y=D^{-1}(g-Cx)). Substitution into the first equation produces
[ (A-BD^{-1}C)x=f-BD^{-1}g. ]
Thus (M/D) is the coefficient matrix governing the variables (x) after the variables (y) have been eliminated. This interpretation is the common structural basis for its appearances in algebra, statistics, and numerical computation.
Inversion and singularity
If (D) and (M/D) are invertible, then (M) is invertible and its inverse has the block form
[ M^{-1}
\begin{pmatrix} S^{-1} & -S^{-1}BD^{-1}\ -D^{-1}CS^{-1} & D^{-1}+D^{-1}CS^{-1}BD^{-1} \end{pmatrix}, \qquad S=A-BD^{-1}C. ]
The corresponding formula with (A) and (M/A) follows by exchanging the two block positions. These formulas identify the inverse of a Schur complement with a principal block of the inverse matrix. In particular,
[ (M^{-1})_{11}=(M/D)^{-1}. ]
Invertibility also separates under the same hypotheses: when (D) is nonsingular, (M) is nonsingular exactly when (M/D) is nonsingular. The determinant identity provides the scalar form of this equivalence, while block elimination provides the operator form.
For singular diagonal blocks, the ordinary definition does not apply because it contains an inverse. Extensions based on the Moore–Penrose inverse define generalized Schur complements, although their properties require additional range conditions. Without those conditions, determinant, rank, and positivity statements need not retain the same form.
Hermitian matrices and inertia
For a Hermitian matrix
[ M= \begin{pmatrix} A & B\ B^* & D \end{pmatrix}, ]
with (D) invertible, a congruence factorization gives
[ M= \begin{pmatrix} I & BD^{-1}\ 0 & I \end{pmatrix} \begin{pmatrix} A-BD^{-1}B^* & 0\ 0 & D \end{pmatrix} \begin{pmatrix} I & 0\ D^{-1}B^* & I \end{pmatrix}. ]
Because congruence preserves inertia, the positive, negative, and zero eigenvalue counts of (M) are the sums of the corresponding counts for (D) and (M/D). This relation is known as the Haynsworth inertia additivity formula.
If (D) is positive definite, then
[
M\succeq 0
\quad\Longleftrightarrow\quad
D\succ0
\ \text{and}
A-BD^{-1}B^*\succeq0.
]
A parallel statement holds with (A) as the eliminated block. The criterion converts positivity of a large block matrix into positivity of one diagonal block and its Schur complement. It underlies many equivalent formulations of linear matrix inequalities.
As a scalar-block example, let
[ M= \begin{pmatrix} A & b\ b^* & d \end{pmatrix}, \qquad d>0. ]
Then (M) is positive semidefinite exactly when
[ A-\frac{1}{d}bb^*\succeq0. ]
The rank-one correction records the contribution of the scalar variable associated with (d).
Probabilistic interpretation
Let a jointly Gaussian random vector be partitioned as
[ \begin{pmatrix} X\ Y \end{pmatrix} \sim \mathcal N\left( \begin{pmatrix} \mu_X\ \mu_Y \end{pmatrix}, \begin{pmatrix} \Sigma_{XX} & \Sigma_{XY}\ \Sigma_{YX} & \Sigma_{YY} \end{pmatrix} \right), ]
where (\Sigma_{YY}) is invertible. The conditional covariance of (X) given (Y) is
[ \operatorname{Cov}(X\mid Y)
\Sigma_{XX} -\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}. ]
This matrix is the Schur complement of (\Sigma_{YY}) in the joint covariance matrix. It measures the residual variation in (X) after the linear dependence associated with (Y) has been accounted for.
The conditional mean is
[ \mathbb E[X\mid Y]
\mu_X+ \Sigma_{XY}\Sigma_{YY}^{-1}(Y-\mu_Y). ]
Both expressions arise from completing the square in the joint Gaussian density. The same block algebra links covariance matrices to precision matrices, where zeros in selected blocks encode conditional independence relations in a Gaussian graphical model.
Optimization and constrained systems
Schur complements occur in systems with equality constraints, including the Karush–Kuhn–Tucker conditions. A typical saddle-point matrix has the form
[ K= \begin{pmatrix} H & G^T\ G & 0 \end{pmatrix}, ]
where (H) represents a Hessian or quadratic form and (G) represents the constraint operator. If (H) is invertible, the Schur complement of (H) is
[ -K/H=-GH^{-1}G^T, ]
with the conventional notation giving
[ K/H=-GH^{-1}G^T. ]
This reduced matrix governs the Lagrange multipliers after the primal variables have been eliminated. Its null space reflects dependencies among the constraints, while its definiteness reflects the behavior of the quadratic form on the relevant constrained subspace.
A related identity converts a quadratic matrix inequality into a block positivity condition. For (A\succ0),
[ C-B^*A^{-1}B\succeq0 ]
is equivalent to
[ \begin{pmatrix} A & B\ B^* & C \end{pmatrix} \succeq0. ]
This equivalence is central to the algebraic representation of semidefinite constraints and to the analysis of convex quadratic expressions.
Numerical interpretation
In numerical linear algebra, a Schur complement is usually represented through a factorization rather than by explicitly forming the inverse appearing in its definition. The notation (BD^{-1}C) denotes the result of solving linear systems with coefficient matrix (D), followed by matrix multiplication. This distinction matters because an explicitly computed inverse can alter sparsity and amplify rounding effects without changing the underlying mathematical expression.
During sparse elimination, eliminating a set of variables creates new couplings among the remaining variables. The matrix encoding those couplings is the Schur complement. Its pattern explains the phenomenon of fill-in in sparse factorizations and connects elimination orderings with graph separators.
The same structure appears in domain-decomposition methods. Variables internal to subdomains are eliminated, while variables on interfaces remain in a reduced system. The resulting interface operator is a Schur complement, and its spectral properties depend jointly on the original operator and the partition of the variables.