In Data-driven stabilization of discrete-time control-affine nonlinear systems: a Koopman operator approach, I read that the following LMI
$$\left(\begin{array}{cccc} \mathbb{U}^{\top} P \mathbb{U}-P & 0 & \mathbb{U}^{\top} P B & k \\ 0 & -P & P B & 0 \\ B^{\top} P \mathbb{U} & B^{\top} P & -\varepsilon P & 0 \\ k^{\top} & 0 & 0 & -\frac{1}{\varepsilon} I \end{array}\right) \prec 0$$
can be reformulated to
$$\left(\begin{array}{ccc} \mathbb{U}^{\top} P \mathbb{U}-P & \mathbb{U}^{\top} P B & k \\ B^{\top} P \mathbb{U} & B^{\top} PB-\varepsilon P & 0 \\ k^{\top} & 0 & -\frac{1}{\varepsilon} I \end{array}\right) \prec 0$$
via the Schur complement. This reformulation is not obvious for me. Can anyone please help me with this reformulation? Any hint would be helpful.
The Schur complement is performed with respect to the 2nd row/column. First note that the matrix inequality is equivalent to
$$\left(\begin{array}{cccc} \mathbb{U}^{\top} P \mathbb{U}-P & \mathbb{U}^{\top} P B & k & 0\\ B^{\top} P \mathbb{U} & -\varepsilon P & 0 & B^{\top} P\\ k^{\top} & 0 & -\frac{1}{\varepsilon} I & 0\\ 0 & P B &0& -P \\ \end{array}\right) \prec 0 .$$
A Schur complement with respect to the last row/column yields
$$\left(\begin{array}{cccc} \mathbb{U}^{\top} P \mathbb{U}-P & \mathbb{U}^{\top} P B & k\\ B^{\top} P \mathbb{U} & -\varepsilon P & 0\\ k^{\top} & 0 & -\frac{1}{\varepsilon} I\\ \end{array}\right)-\begin{pmatrix}0\\B^\top P\\0\end{pmatrix}(-P)^{-1}\begin{pmatrix}0 & P B &0\end{pmatrix} \prec 0$$
from which the result follows.
As a matter of fact, it is always easier (at least at the beginning) to permute rows and columns you want to perform a Schur complement with at the end of the matrix and then use the standard formula.