Schur Complement

Consider a block matrix of the form

M = [\begin{matrix} A & B \\ C & D \end{matrix}],

where the diagonal matrices are $p \times p$ and $q \times q$ . If $A$ is invertible, then the Schur complement of the block $A$ of $M$ is defined as

M / A = D - C A^{- 1} B .

This term comes up when solving a system of linear equations involving $M$ :

[\begin{matrix} A & B \\ C & D \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} u \\ v \end{matrix}] .

Taking the first row and the rearranging, we have

\begin{matrix} A x + B y = u, \\ x = A^{- 1} (u - B y) . \end{matrix}

Then plugging this into the second row and simplifying,

\begin{matrix} C x + D y = v, \\ C A^{- 1} u - C A^{- 1} B y + D y = v, \\ \underset{M / A}{\underset{⏟}{(D - C A^{- 1} B)}} y = v - C A^{- 1} u . \end{matrix}

This is the reduced equation, as $x$ has been eliminated from the system.

In other words, the Schur complement appears when performing block Gaussian elimination, E.g.

[\begin{matrix} A & B \\ C & D \end{matrix}] \to [\begin{matrix} A & B \\ C & D \end{matrix}] [\begin{matrix} I_{p} & 0 \\ - D^{- 1} C & I_{q} \end{matrix}] = [\begin{matrix} A - B D^{- 1} C & B \\ 0 & D \end{matrix}] .