Schur Complement - Background

Background

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 Ip 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

\begin{align} ML &= \left\left = \left \\ &= \left \left. \end{align}

That is, we have shown that

\begin{align} \left &= \left \left \left, \end{align}

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

\begin{align} & {} \quad \left^{-1} = \left \left \left \\ & = \left. \end{align}

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

If M is a positive-definite symmetric matrix, then so is the Schur complement of D in M.

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.

Read more about this topic:  Schur Complement

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