Schur complement

From Academic Kids

In linear algebra and the theory of matrices, the Schur complement (named after Issai Schur) 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

<math>M=\left[\begin{matrix} A & B \\ C & D \end{matrix}\right]<math>

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

<math>A-BD^{-1}C.<math>

The Schur complement arises as the result of performing a "partial" Gaussian elimination by multiplying the matrix M from the right with the "lower triangular" block matrix

<math>LT=\left[\begin{matrix} E_p & 0 \\ -D^{-1}C & D^{-1} \end{matrix}\right].<math>

Here Ep denotes a p×p unit matrix. After multiplication with the matrix LT the Schur complement appears in the upper p×p block. The product matrix is

<math>M\cdot LT=\left[\begin{matrix} A-BD^{-1}C & BD^{-1} \\ 0 & E_q \end{matrix}\right].<math>

If M is a positive definite symmetric matrix, 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) in 'Rn+m has a multivariate normal distribution whose variance is the symmetric positive-definite matrix

<math>V=\left[\begin{matrix} A & B \\ B^T & C \end{matrix}\right].<math>

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

<math>\operatorname{var}(X\mid Y)=A-BC^{-1}B^T.<math>

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.it:Complemento di Schur

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