The Schur complement is a key tool in the fields of linear algebra, the theory of matrices, numerical analysis, and statistics.
It is defined for a block matrix.
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.
[2] Emilie Virginia Haynsworth was the first to call it the Schur complement.
[3] The Schur complement is sometimes referred to as the Feshbach map after a physicist Herman Feshbach.
[4] The Schur complement arises when performing a block Gaussian elimination on the matrix M. In order to eliminate the elements below the block diagonal, one multiplies the matrix M by a block lower triangular matrix on the right as follows:
where Ip denotes a p×p identity matrix.
As a result, the Schur complement
leads to an LDU decomposition of M, which reads
Thus, the inverse of M may be expressed involving D−1 and the inverse of Schur's complement, assuming it exists, as
The above relationship comes from the elimination operations that involve D−1 and M/D.
An equivalent derivation can be done with the roles of A and D interchanged.
By equating the expressions for M−1 obtained in these two different ways, one can establish the matrix inversion lemma, which relates the two Schur complements of M: M/D and M/A (see "Derivation from LDU decomposition" in Woodbury matrix identity § Alternative proofs).
The Schur complement arises naturally in solving a system of linear equations such as[7]
Substituting this expression into the second equation yields We refer to this as the reduced equation obtained by eliminating
The matrix appearing in the reduced equation is called the Schur complement of the first block
: Solving the reduced equation, we obtain Substituting this into the first equation yields We can express the above two equation as: Therefore, a formulation for the inverse of a block matrix is: In particular, we see that the Schur complement is the inverse of the
block entry of the inverse of
to be well-conditioned in order for this algorithm to be numerically accurate.
This method is useful in electrical engineering to reduce the dimension of a network's equations.
is zero, we can eliminate the associated rows of the coefficient matrix without any changes to the rest of the output vector.
, thus reducing the dimension of the coefficient matrix while leaving
This is used to advantage in electrical engineering where it is referred to as node elimination or Kron reduction.
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 covariance is the symmetric positive-definite matrix where
Then the conditional covariance of X given Y is the Schur complement of C in
In that case, the Schur complement of C in
[citation needed] Let X be a symmetric matrix of real numbers given by
Then The first and third statements can be derived[7] by considering the minimizer of the quantity
and similarly for positive semi-definite matrices, the second (respectively fourth) statement is immediate from the first (resp.
There is also a sufficient and necessary condition for the positive semi-definiteness of X in terms of a generalized Schur complement.