# Schur complement

In linear algebra and the theory of matrices, the **Schur complement** of a block 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 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.^{[1]} Emilie Virginia Haynsworth was the first to call it the *Schur complement*.^{[2]} The Schur complement is a key tool in the fields of numerical analysis, statistics and matrix analysis.

## BackgroundEdit

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

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

## PropertiesEdit

- 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.

- In general, if
*A*is invertible, then

- whenever this inverse exists.

- The determinant of
*M*is also clearly seen to be given by

- which generalizes the determinant formula for 2 × 2 matrices.

- (Guttman rank additivity formula) The rank of
*M*is given by - (Haynsworth inertia additivity formula) The
*inertia*of the block matrix*M*is equal to the inertia of*A*plus the inertia of*M*/*A*.

## Application to solving linear equationsEdit

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 statisticsEdit

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 ^{[3]}:

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.^{[citation needed]}

## Schur complement condition for positive definiteness and positive semi-definitenessEdit

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.,

Then

*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^{[4]} 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) statement.

There is also a sufficient and necessary condition for the positive semi-definiteness of *X* in terms of a generalized Schur complement.^{[1]} Precisely,

- and

where denotes the generalized inverse of .

## See alsoEdit

## ReferencesEdit

- ^
^{a}^{b}Zhang, Fuzhen (2005).*The Schur Complement and Its Applications*. Springer. doi:10.1007/b105056. ISBN 0-387-24271-6. **^**Haynsworth, E. V., "On the Schur Complement",*Basel Mathematical Notes*, #BNB 20, 17 pages, June 1968.**^**von Mises, Richard (1964). "Chapter VIII.9.3".*Mathematical theory of probability and statistics*. Academic Press. ISBN 978-1483255385.**^**Boyd, S. and Vandenberghe, L. (2004), "Convex Optimization", Cambridge University Press (Appendix A.5.5)