# Conditions for a block matrix to be Hurwitz stable

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?

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.
• How do you compute the eigenvalues of $A$?
• Look at $A \pmatrix{u\cr v\cr} - \lambda \pmatrix{u\cr v\cr}$ explicitly. Oct 20, 2021 at 14:08