TLDR
I need to use what looks like the Schur complement to transform a linear matrix inequality but instead of a $2\times 2$ block matrix there are more blocks.
Question
I'm having trouble with a derivation in this paper on Risk-averse Model Predictive Control. The part I am struggling with is equation (16) where the method proposed by the paper is being applied to a practical example - a Markov Jump Linear system. A real understanding of the topic is, however, not necassary to help I think. I just can't follow a specific step in a derivation.
It goes as follows:
The author starts with
$$Q_i + K_i^TR_iK_i+\sum_{j\in cov(i)}\mu_{ij}(A_i+B_iK_i)^TP_j^f(A_i+B_iK_i)\preceq P_i^f$$
where $Q_i$ is positive semidefinite and $R_i$ is positive definite. The notation $cov(i)$ denotes a set of integers representing the possible next states. If a certain integer $k\notin cov(i)$, then $\mu_{ik}=0$. There are $n$ possible states.
By a change of variables $(P_i^f)^{-1} = M_i$, $K_i = Y_iM_i^{-1}$, $F_i = \left[\sqrt{\mu_{i1}}I \cdots\sqrt{\mu_{in}}I\right]$ and $M = \mathbf{blkdiag}(M_1 \cdots M_n)$ the inequality is transformed into the following linear matrix inequality (LMI):
$$ \begin{bmatrix} M_i & (A_iM_i+B_iY_i)^TF_i & M_iQ_i^{1/2} & Y_iR_i^{1/2}\\ \ast& M& 0& 0\\ \ast& \ast& I& 0\\ \ast& \ast& \ast& I\\ \end{bmatrix} \succeq 0 $$
The matrix in above inequality is symmetric. It is getting to this matrix inequality which I am having trouble with.
I started by substituting the new variables and by summing over all possible $j$.
$$Q_i + M_i^{-T}Y_i^TR_iY_iM_i^{-1} + \sum_{j=1}^n\mu_{ij}(A_i+B_iY_iM_i^{-1})^T M_j^{-1}(A_i+B_iY_iM_i^{-1})\preceq M_i^{-1}$$
Next, by multiplying left with $M_i^T$ and right with $M_i$ I got: $$\implies M_i^TQ_iM_i + Y_i^TR_iY_i + \sum_{j=1}^n\mu_{ij}(A_iM_i+B_iY_i)^T M_j^{-1}(A_iM_i+B_iY_i)\preceq M_i^T$$
Any help on how to proceed to get to the linear matrix inequality would be much appreciated.
EDIT
I just learned about the Schur complement, which seems like it might be useful here but I don't know how to adapt it to make it applicable in this situation.