Schur complement

In linear algebra and the theory of matrices, the Schur complement of a matrix block (i.e., a submatrix within a larger matrix) is defined as follows. Suppose A, B, C, D are respectively p×p, p×q, q×p and q×q matrices, and D is invertible. Let

so that M is a (p+q)×(p+q) matrix.

Then the Schur complement of the block D of the matrix M is the p×p matrix

and the Schur complement of the block A of the matrix M is the q×q matrix

In the case that A or D is singular, the inverses on M/A and M/D can be replaced by a generalized inverse, yielding what is called 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 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 the "block lower triangular" matrix

Here I_p denotes a p×p identity matrix. After multiplication with the matrix L the Schur complement appears in the upper p×p block. The product matrix is

This is analogous to an LDU decomposition. That is, we have shown that

and inverse of M thus may be expressed involving D⁻¹ and the inverse of Schur's complement (if it exists) only as

C.f. matrix inversion lemma which illustrates relationships between the above and the equivalent derivation with the roles of A and D interchanged.

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, and A, B, C, D are as above. Multiplying the bottom equation by ${\displaystyle BD^{-1}}$ and then subtracting from the top equation one obtains

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