List of Taylor polyhedra

The following identity holds for a Schur complement of a square matrix:

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

$L=\left[\begin{array}{cc}{I}_p& 0\\ -D^{-1}C& I_q\end{array}\right].$

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

$\begin{array}{rl}ML& =\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\left[\begin{array}{cc}{I}_p& 0\\ -D^{-1}C& I_q\end{array}\right]=\left[\begin{array}{cc}A-B{D}^{-1}C& B\\ 0& D\end{array}\right]\\ & =\left[\begin{array}{cc}{I}_p& B{D}^{-1}\\ 0& I_q\end{array}\right]\left[\begin{array}{cc}A-B{D}^{-1}C& 0\\ 0& D\end{array}\right].\end{array}$

That is, we have shown that

$\begin{array}{rl}\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]& =\left[\begin{array}{cc}{I}_p& B{D}^{-1}\\ 0& I_q\end{array}\right]\left[\begin{array}{cc}A-B{D}^{-1}C& 0\\ 0& D\end{array}\right]\left[\begin{array}{cc}{I}_p& 0\\ D^{-1}C& I_q\end{array}\right],\end{array}$