Schur complement

In linear algebra and the theory of matrices, the Schur complement of a block of a matrix within the larger matrix is defined as follows. 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

If M is positive definite, then so is the Schur complement of D in M.

Applications to probability theory

Suppose the random column vectors X, Y live in Rⁿ and R^m respectively, and the vector (X′, Y′)′ (where a′ = the transpose of a) has a multivariate normal distribution whose variance is the symmetric positive-definite matrix

Then the conditional variance of X given Y is the Schur complement of C in V:

If we take the matrix V above to be, not a variance of a random vector, but a sample variance, then it may have a Wishart distribution. In that case, the Schur complement of C in V also has a Wishart distribution.