Is this block matrix Hurwitz?

Question

Let $A\in\mathbb{R}^{n\times n}$ be Hurwitz. Let $k_1,k_2>0$. Consider the matrix

$$ M = \begin{bmatrix} 0 & I\\ k_1 A & k_2 A \end{bmatrix}, $$

where $0$ and $I$ denote respectively the zero matrix and the identity matrix in $\mathbb{R}^{n\times n}$.

1) Is $M$ also Hurwitz for any $k_1,k_2>0$?

2) If not, is there some necessary and/or sufficient condition on $k_1$ and $k_2$ for $M$ to be Hurwitz?

Edit: fixed a wrong sign

Answer 1

For sake of having an answer, note that if $z$ is an eigenvalue of $A$, then the roots of $t(t-k_2z)-k_1z=0$, i.e. $$ \frac{k_2z\pm\sqrt{k_2^2z^2+4k_1z}}{2}, $$ would be eigenvalues of $M$. Therefore,

$M$ is Hurwitz if and only if $\operatorname{Re}(z+\sqrt{z^2+pz})<0$ for every eigenvalue $z$ of $A$ and every square root $\sqrt{z^2+pz}$ of $z^2+pz$, where $p=\frac{4k_1}{k_2^2}>0$.

Unfortunately, this necessary and sufficient condition is not always satisfied. E.g. when $z=-1+3i$ and $p=8$, we have $\operatorname{Re}(z+\sqrt{z^2+pz})=1.0104$ for one of the square roots. This gives us a counterexample for the OP: $$ k_1=2,\ k_2=1,\ A=\pmatrix{-1&-3\\ 3&-1}. $$ The eigenvalues of $M$ are numerically $\color{red}{0.50519\pm3.73839i}$ and $-1.50519\pm0.73839i$.