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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 as the result of performing a block Gaussian elimination by multiplying the matrix M from the right with a block lower triangular matrix
Here Ip denotes a p×pidentity matrix. After multiplication with the matrix L the Schur complement appears in the upper p×p block. The product matrix is
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, A, B, C, D are as above, and D is invertible. 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 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
where is the covariance matrix of X, is the covariance matrix of Y and is the covariance matrix between X and Y.
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-definiteness
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: