Schur complement

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In linear algebra and the theory of matrices, the Schur complement of a block matrix is defined as follows.

Suppose p, q are nonnegative integers, and suppose A, B, C, D are respectively p × p, p × q, q × p, and q × q matrices of complex numbers. Let

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 matrix defined by

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.

Background[edit]

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 appears in the upper-left p×p block.

Continuing the elimination process beyond this point (i.e., performing a block Gauss–Jordan elimination),

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).

Properties[edit]

  • 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.
  • 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.

Application to solving linear equations[edit]

The Schur complement arises naturally in solving a system of linear equations such as[3]

.

Assuming that the submatrix is invertible, we can eliminate from the equations, as follows.

.

Substituting this expression into the second equation yields

.

We refer to this as the reduced equation obtained by eliminating from the original equation. The matrix appearing in the reduced equation is called the Schur complement of the first block in :

.

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 .

In practice, one needs 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 statistics[edit]

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 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 :[4]

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]

Conditions for positive definiteness and semi-definiteness[edit]

Let X be a symmetric matrix of real numbers given by

Then

  • If A is invertible, then X is positive definite if and only if A and its complement X/A are both positive definite:
    [5]
  • 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:
    [5]
  • 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 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) 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 also[edit]

References[edit]

  1. ^ a b Zhang, Fuzhen (2005). Zhang, Fuzhen (ed.). The Schur Complement and Its Applications. Numerical Methods and Algorithms. Vol. 4. Springer. doi:10.1007/b105056. ISBN 0-387-24271-6.
  2. ^ Haynsworth, E. V., "On the Schur Complement", Basel Mathematical Notes, #BNB 20, 17 pages, June 1968.
  3. ^ a b Boyd, S. and Vandenberghe, L. (2004), "Convex Optimization", Cambridge University Press (Appendix A.5.5)
  4. ^ von Mises, Richard (1964). "Chapter VIII.9.3". Mathematical theory of probability and statistics. Academic Press. ISBN 978-1483255385.
  5. ^ a b Zhang, Fuzhen (2005). The Schur Complement and Its Applications. Springer. p. 34.