Given the parameter dependent matrix $A=\begin{pmatrix} 0 & I\\ A_1 & kA_2\end{pmatrix}$, with $A_1, A_2\in \mathbb{R}^{n\times n}$ and $k\in\mathbb{R}>0$, is there a way to display the eigenvalues of $A$ as a function of $A_1, A_2, k$? Or to give an estimation of the eigenvalues' location? What can be said about the corresponding eigenvectors of $A$ dependent on $k$?
We can suppose that $A_1,A_2$ have full rank and all eigenvalues are negative or have a negative real part. They may have repeated eigenvalues.
Clearly if $\lambda(A)$ is an eigenvalue, for $k=0$ we have $\lambda(A)=\pm \sqrt{\lambda(-A_1)}$, and for $k\rightarrow \infty$ we have $n$ eigenvalues $\lambda(A)=0$ and $n$ eigenvalues $\lambda(A)=\lambda(kA_2)$. Furthermore we have $k\mathrm{tr}(B)=\sum_{i=1}^{2n} \lambda_i(A)$. Assuming that $\mathrm{tr}(B)<0$ this would mean that at least some eigenvalues of $A$ have negative real parts of magnitude growing with $k$.
I'm interested in what happens for $0\ll k\ll\infty$. My intuition is that with growing $k$ the eigenvalues of $A$ will move into the left half-plane, before tending to $-\infty$ (as all eigenvalues of $A_2$ have negative real parts) or $0$ but so far I haven't been able to find a suitable proof method for this.