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.Emilie Virginia Haynsworth was the first to call it the Schur complement. The Schur complement is a key tool in the fields of numerical analysis, statistics, and matrix analysis.
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×pidentity matrix. As a result, the Schur complement appears in the upper-left p×p block.
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).
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.
(Schur's formula) When A, respectively D, is invertible, 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) If D is invertible, then the rank of M is given by
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.
Conditions for positive definiteness and semi-definitenessEdit
Let X be a symmetric matrix of real numbers given by
If A is invertible, then X is positive definite if and only if A and its complement X/A are both positive definite:
^ abZhang, 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.
^Haynsworth, E. V., "On the Schur Complement", Basel Mathematical Notes, #BNB 20, 17 pages, June 1968.
^Crabtree, Douglas E.; Haynsworth, Emilie V. (1969). "An identity for the Schur complement of a matrix". Proceedings of the American Mathematical Society. 22 (2): 364–366. doi:10.1090/S0002-9939-1969-0255573-1. ISSN 0002-9939. S2CID 122868483.
^Devriendt, Karel (2022). "Effective resistance is more than distance: Laplacians, Simplices and the Schur complement". Linear Algebra and Its Applications. 639: 24–49. arXiv:2010.04521. doi:10.1016/j.laa.2022.01.002. S2CID 222272289.
^ abBoyd, S. and Vandenberghe, L. (2004), "Convex Optimization", Cambridge University Press (Appendix A.5.5)
^von Mises, Richard (1964). "Chapter VIII.9.3". Mathematical theory of probability and statistics. Academic Press. ISBN 978-1483255385.
^ abZhang, Fuzhen (2005). The Schur Complement and Its Applications. Springer. p. 34.