How to prove that a certain block matrix is positive semi definite, which depends on a undetermined submatrix

How should I proof the following matrix $$M = \begin{pmatrix} Z-A^TZA & -A^TZB\\ -B^TZA & -B^TZB \end{pmatrix},$$ to be positive semidefinite? The matrices $A\in \mathbb{R}^{n\times n}$ and $ B\in \mathbb{R}^{n\times m} $ are known. The matrix $Z\in R^{n\times n}$ is unknown and the actual goal is to construct the matrix $Z$ such that $M$ is positive semi-definite.

I have tried the following things:

• constructing $Z$ as a diagonal matrix and by using a symbolic programming library (sympy) I found the expressions for the eigenvalues of $M$. Here I found out that for this specific $Z$, the eigenvalues will be indefinite (larger or smaller than zero)
• using theories off Schur Complement: Wikipedia Link. For example, in order for the third option of the link to be true; $Z-A^TZA \succ 0$ and $ -B^TZB-(-B^TZA)^T(Z-A^TZA)^{-1}(-A^TZB) \succeq 0$, $(-B^TZA)^T= -A^TZB$ -> $Z$ should be symmetric.

I think I could write this all out to be a larger Linear Matrix Inequality (LMI) and try to solve using python for example. However, I'm not sure if that is true and if this is the correct approach as this would take some time to implement this in a python code.

Do you have any suggestions or feedback for a different approach?

(My mathematical knowledge is not so great as I come from a mechanical/control engineering background. Furthermore, this is my first question on a stack website. I apologize for any trivial mistakes I made)

EDIT:

Extra info regarding the matrices $A$,$B$; they are so-called state-space matrices which are part of a Linear Time-Invariant(LTI) dynamical model Wikipedia. The standard form is $x_{k+1}=Ax_k + Bu$ where $x \in \mathbb{R}^n$ represents the state of the system and $u \in \mathbb{R}^m$ is the input to this dynamical model. The matrix $A$ is the mapping of the current state of the system to the next state (in time) and $B$ represents the mapping of the input to the next state (in time). These state-space matrices ($A, B$) are thus known as they represent the dynamics of a system. However, they have no special mathematical properties.

Solution 1:

This will not be a definite answer, but I think it can help:

Let's assume $Z$ is symmetric, and consider a general vector $v \in \mathbb{R}^{n+m}$ decomposed as $v^T = (v_1^T , v_2^T)$ suitably to match matrix block dimensions.

Now for $M$ to be spd, we need $v^T M v \geq 0 \ \forall v \in \mathbb{R}^{n+m}$. If we develop the expression, we get

$\begin{equation} 0 \leq v_1^T Z v_1 - v_1^T A^T Z A v_1 - v_1^T A^T Z B v_2 - v_2^T B^T Z A v_1 - v_2^T B^T Z B v_2 \end{equation}$

and taking advantage that $Z$ is symmetric and regrouping terms we have that

$0 \leq v_1^T Z v_1 - \left(v_1 A + v_2 B\right)^T Z \left(v_1 A + v_2 B \right)$

or even

$0 \leq \left(v_1 + v_1 A - v_2 B\right)^T Z \left(v_1 - v_1 A - v_2 B \right)$

which is quite more manageable.

Cheers!