Schur Complement¶
We can perform elimination by blocs for a matrix M:
\[\begin{split} M = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \end{split}\]
We want to get \(0\) below A this can be achieved with:
\[\begin{split}\begin{bmatrix} I & 0 \\ -CA^{-1} & I \end{bmatrix} \begin{bmatrix} A && B \\ C && D \end{bmatrix} = \begin{bmatrix} A && B \\ 0 && D - CA^{-1}B \end{bmatrix} \end{split}\]
Where \(D - CA^{-1}B\) is called Schur complement also denoted \(M / A\)