Could someone please help?
The question reads: For which real numbers $k$ is the zero state a stable equilibrium of the dynamic system $x_{t+1} = Ax_t$?
$A = \begin{bmatrix} 0.1 &k \\ 0.3 & 0.3 \end{bmatrix}$
So, my thought is I need to find the eigenvalues. In order to do this I calculated the characteristic polynomial as
$x^2 - 0.4x + 0.03 - 0.3k = 0$, with $x$ representing eigenvalues.
Using the quadratic formula I found that the (real) eigenvalues are $x = \frac{2 \pm \sqrt{1+30k}}{10}$ and for the zero state to be in stable equilibrium $\sqrt{1+30k}<8$. Hence, $k<21/10$ (for stable equilibrium).
My question is how do I figure out the values for $k$ if the eigenvalues are complex?
Do I solve the inequality $\sqrt{-1-30k} < 8$?
Thanks!