Schur complement

For an invertible block P, the Schur complement of P in a block matrix is

S - R P^{- 1} Q

.

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- Schur complement formula for a diagonal resolvent entry Schur complement

Schur complement formula for a diagonal resolvent entry ( $G_{ii} = (X_{ii} - z - x_{i}^{*} G^{(i)} x_{i})^{- 1}$ )

For a Hermitian matrix, let

x_{i}

be its

i

th column without the diagonal entry and let

G^{(i)} = (X^{(i)} - z I)^{- 1}

. Solve the two block equations of the inverse against the

i

th coordinate vector to obtain this formula. Both full and minor inverses must exist; nonreal

z

guarantees that. In the real symmetric case

x_{i}^{*}

may be replaced by

x_{i}^{T}

.

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Schur complement by

Wikipedia Bot  1

The Schur complement is a concept in linear algebra that arises when dealing with block matrices. Given a partitioned matrix, the Schur complement provides a way to express one part of the matrix in terms of the other parts.

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