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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!

  • 0
    Just so you know, I reformatted your question using LaTeX for easier viewing.2011-01-06
  • 1
    For the real case, you also need $k\geq \cfrac{-1}{30}$ to keep them real.2011-01-06
  • 0
    You're quite welcome. Glad it was helpful.2011-01-06

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