Consider the following block matrix:
$$
A = \begin{bmatrix}
0 & I\\
-M & -I
\end{bmatrix}
$$
Suppose matrix $M$ is positive definite and satisfies $M\succeq \alpha I$, where $\alpha>0$ is a constant. When will matrix $A$ be Hurwitz stable, i.e., all of the eigenvalues have negative real parts?



Conditions for a block matrix to be Hurwitz stable
The eigenvalues of $A$ are $(-1 \pm \sqrt{1-4s})/2$ where $s$ is an eigenvalue of $M$. If $s \ge 1/4$, these have real part $-1/2$, while if $0 < s < 1/4$, they are both real and negative. So it's always true when $M$ is positive definite.


