Suppose A, B, C, D are respectively p × p, p × q, q × p, and q × q matrices, and D is invertible. Let
so that M is a (p + q) × (p + q) matrix.
Then the Schur complement of the block D of the matrix M is the p × p matrix defined by
and, if A is invertible, the Schur complement of the block A of the matrix M is the q × q matrix defined by
In the case that A or D is singular, substituting a generalized inverse for the inverses on M/A and M/D yields the generalized Schur complement.
The Schur complement is named after Issai Schur who used it to prove Schur's lemma, although it had been used previously. Emilie Haynsworth was the first to call it the Schur complement. The Schur complement is a key tool in the fields of numerical analysis, statistics and matrix analysis.
The Schur complement arises naturally in solving a system of linear equations such as
where x, a are p-dimensional column vectors, y, b are q-dimensional column vectors, and A, B, C, D are as above. Multiplying the bottom equation by and then subtracting from the top equation one obtains
Thus if one can invert D as well as the Schur complement of D, one can solve for x, and
then by using the equation one can solve for y. This reduces the problem of inverting a matrix to that of inverting a p × p matrix and a q × q matrix. In practice, one needs D to be well-conditioned in order for this algorithm to be numerically accurate.
In electrical engineering this is often referred to as node elimination or Kron reduction.
Applications to probability theory and statistics
Suppose the random column vectors X, Y live in Rn and Rm respectively, and the vector (X, Y) in Rn + m has a multivariate normal distribution whose covariance is the symmetric positive-definite matrix
where is the covariance matrix of X, is the covariance matrix of Y and is the covariance matrix between X and Y.
If we take the matrix above to be, not a covariance of a random vector, but a sample covariance, then it may have a Wishart distribution. In that case, the Schur complement of C in also has a Wishart distribution.
Schur complement condition for positive definiteness and positive semi-definiteness
Let X be a symmetric matrix given by
Let X/A be the Schur complement of A in X; i.e.,
and X/C be the Schur complement of C in X; i.e.,
X is positive definite if and only if A and X/A are both positive definite:
X is positive definite if and only if C and X/C are both positive definite:
If A is positive definite, then X is positive semi-definite if and only if X/A is positive semi-definite:
If C is positive definite, then X is positive semi-definite if and only if X/C is positive semi-definite:
The first and third statements can be derived by considering the minimizer of the quantity
as a function of v (for fixed u).
and similarly for positive semi-definite matrices, the second (respectively fourth) statement is immediate from the first (resp. third) statement.
There is also a sufficient and necessary condition for the positive semi-definiteness of X in terms of a generalized Schur complement. Precisely,