α and ω possible limit sets of points

Question

1. What are all the $α$ and $ω$ possible limit sets of points for:

$$A=\begin{pmatrix}-4&-2\\3&-11 \end{pmatrix}.$$

I am not really sure what to do..

$$\dot{x}=\begin{pmatrix}-4&-2\\3&-11 \end{pmatrix}x$$

Answer 1

The eigenvalues of $A$ are $-5$ and $-10$, both negative, hence:

• For every "initial" condition $x(0)$, $x(t)\to0$ when $t\to+\infty$
• For every "initial" condition $x(0)\ne0$, $\|x(t)\|\to+\infty$ when $t\to-\infty$