# Schur complement

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In linear algebra and the theory of matrices,
the **Schur complement** of a matrix block (i.e., a submatrix within a
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

It is named after Issai Schur who used it to prove Schur's lemma, although it had been used previously.^{[1]} Emilie Haynsworth was the first to call it the *Schur complement*.^{[2]}

## Background

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

Here *I _{p}* denotes a

*p*×

*p*identity matrix. After multiplication with the matrix

*L*the Schur complement appears in the upper

*p*×

*p*block. The product matrix is

This is analogous to an LDU decomposition. That is, we have shown that

and inverse of *M* thus may be expressed involving *D*^{−1} and the inverse of Schur's complement (if it exists) only as

C.f. matrix inversion lemma which illustrates relationships between the above and the equivalent derivation with the roles of *A* and *D* interchanged.

If *M* is a positive-definite symmetric matrix, then so is the Schur complement of *D* in *M*.

If *p* and *q* are both 1 (i.e. *A*, *B*, *C* and *D* are all scalars), we get the familiar formula for the inverse of a 2-by-2 matrix:

provided that *AD* − *BC* is non-zero.

Moreover, the determinant of *M* is also clearly seen to be given by

which generalizes the determinant formula for 2x2 matrices.

## Application to solving linear equations

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.

## Applications to probability theory and statistics

Suppose the random column vectors *X*, *Y* live in **R**^{n} and **R**^{m} respectively, and the vector (*X*, *Y*) in **R**^{n+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*.

Then the conditional covariance of *X* given *Y* is the Schur complement of *C* in :

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.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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## Schur complement condition for positive definiteness

Let *X* be a symmetric matrix given by

Let *S* be the Schur complement of *A* in *X*, that is:

Then

The first and third statements can be derived^{[3]} by considering the minimizer of the quantity

as a function of *v* (for fixed *u*).

Furthermore, since

and similarly for positive semi-definite matrices, the second (respectively fourth) statement is immediate from the first (resp. third).

## See also

- Woodbury matrix identity
- Quasi-Newton method
- Haynsworth inertia additivity formula
- Gaussian process
- Total least squares