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Wikipedia / Schur complement

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. Suppose p, q are nonnegative integers such that p + q > 0, and suppose A, B, C, D are respectively p × p, p × q, q × p, and q × q so that M is a (p + q) × (p + q) matrix.

If D is invertible, then the Schur complement of the block D of the matrix M is the p × p If A is invertible, the Schur complement of the block A of the matrix M is the q × q 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. Emilie Virginia Haynsworth was the first to call it the Schur complement. The Schur complement is sometimes referred to as the Feshbach map after a physicist Herman Feshbach.

Background

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 where Ip denotes a pp identity matrix appears in the upper-left pp block.

Continuing the elimination process beyond this point (i.e., performing a block Gauss–Jordan elimination leads to an LDU decomposition of M Thus, the inverse of M may be expressed involving D1 The above relationship comes from the elimination operations that involve D1 and M/D. An equivalent derivation can be done with the roles of A and D interchanged. By equating the expressions for M1 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 ).

Properties

  • 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 ADBC is non-zero.

  • In general, if A is invertible, then

  • whenever this inverse exists.

  • (Schur's formula) When A, respectively D, is invertible, the determinant of M is also clearly seen to be given by

  • , respectively

  • ,

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

  • (Guttman rank additivity formula) If D is invertible, then the rank of M is given by

    (Haynsworth inertia additivity formula) If A is invertible, then the inertia of the block matrix M is equal to the inertia of A plus the inertia of M/A.

  • .

  • The Schur complement of a Laplacian matrix is also a Laplacian matrix.

Application to solving linear equations

The Schur complement arises naturally in solving a system of linear equations such as

.

from the equations, as follows.

Substituting this expression into the second equation yields

We refer to this as the reduced equation:

  1. .

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:

.

to be well-conditioned in order for this algorithm to be numerically accurate.

is zero, we can eliminate the associated rows of the coefficient matrix unmodified. This is used to advantage in electrical engineering where it is referred to as node elimination or Kron reduction.

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 covariance is the symmetric positive-definite matrix

is the covariance matrix of X is the covariance matrix of Y is the covariance matrix between X and Y.

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

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 also has a Wishart distribution.

Conditions for positive definiteness and semi-definiteness

Let X be a symmetric matrix Then by the Haynsworth inertia additivity formula, we find

  • If A is invertible, then X is positive definite if and only if A and its complement X/A are both positive definite:

    If C is invertible, then X is positive definite if and only if C and its complement X/C are both positive definite:

    If A is positive definite, then X is positive semi-definite if and only if the complement X/A is positive semi-definite:

    If C is positive definite, then X is positive semi-definite if and only if the complement X/C is positive semi-definite:

The first and third statements can also be derived as a function of v (for fixed u).

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. Precisely,

  • and

denotes a generalized inverse.

See also

References