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 and statistics
Suppose the random column vectors X, Y live in Rn and Rm 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.
Referenced By
List of linear algebra topics | List of mathematical topics (S-U) | Woodbury's identity | Woodbury matrix identity
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