For a matrix $A$, its dimension is $N$ * $N (N=1,2,...)$.
The characteristic equation of the matrix A is:
$(x^2+a_2 x+a_1)^N-(a_3 x+a_1)^N-k x (x^2+a_2 x+a_1)^{(N-1)}=0$,
where $x$ is the eigenvalue, and $a_1$, $a_2$, $a_3$ are pre-defined parameters. $a_2^2-a_3^2-2 a_1<0, a_1>0, a_2>a_3>0$.
If $\frac{-a_2\pm\sqrt{a_2^2-4 a_1}}{2}$ is not equal to $\frac{-a_1}{a_3}$, then they are not the eigenvalues, and the characteristic equation can be divided by $(x^2+a_2 x+a_1)^N$, and then the following equation can be obtained.
$1-{(\frac{a_3 x+a_1}{x^2+a_2 x+a_1})}^N-\frac{k x}{x^2+a_2 x+a_1}= 0$
Problem:
$k$ is nonpositive real number. What I want to do is to detemine $k$ to make all the eigenvalues have non-positive real parts. Can we find a condition or kind of approximation between $k$ and $N$, $a_1$, $a_2$, $a_3$?