Given a square symmetric matrix $H\in\mathbb{R}^{n\times n}$, design a symmetric positive definite matrix $M\in\mathbb{R}^{n\times n}$ and positive scalar $\alpha$ such that the following ${3n\times 3n}$ matrix is Schur stable (all eigenvalues in open unit disk):

$$ A=\begin{bmatrix} I & 0&-\alpha I\\ I&0&0\\ (M+H)^{-1} & 0 &(M+H)^{-1}(M-2\alpha I) \end{bmatrix} $$ where $I$ is the ${n\times n}$ Identity matrix. This problem originates from the stability analysis of a discrete-time linear system. I tried numerical examples and it is easy to find such $M$ and $\alpha$.

I have no idea how to design such $M$ in general or deduct such requirements on $M$.

One thought is that construct a symmetric positive definite $P$ such that $A^\top P A \prec P$. But the freedom of $P$ is so large that I do not know where to start.

Given a square symmetric matrix $H\in\mathbb{R}^{n\times n}$, design a symmetric positive definite matrix $M\in\mathbb{R}^{n\times n}$ and positive scalar $\alpha$ such that the following ${3n\times 3n}$ matrix is Schur stable (all eigenvalues in open unit disk):

$$ A=\begin{bmatrix} I & 0&-\alpha I\\ I&0&0\\ (M+H)^{-1} & 0 &(M+H)^{-1}(M-2\alpha I) \end{bmatrix} $$ where $I$ is the ${n\times n}$ Identity matrix. This problem originates from the stability analysis of a discrete-time linear system. I tried numerical examples and it is easy to find such $M$ and $\alpha$.

I have no idea how to design such $M$ in general or deduce such requirements on $M$.

One thought is that construct a symmetric positive definite $P$ such that $A^\top P A \prec P$. But the freedom of $P$ is so large that I do not know where to start.