I got stuck with a problem while studying for a control systems exam. It goes as following:
"Look at the picture of trajectories of a linear, time-invariant system with the form: $\frac{d\mathbf{x}}{dt}=\mathbf{A}x$.
The Eigenvalues of the matrix $A$ are $s_1=-1$ and $s_2=-2$.
Find the Eigenvectors $p_1$ and $p_2$ considering the given Eigenvalues."
How can I calculate the Eigenvectors with just knowing the Eigenvalues and the trajectories?
Thanks in advance
Hint:
The general solution is $$ x(t) = c_1 \xi_1 e^{\lambda_1 t} + c_2 \xi_2 e^{\lambda_2 t}, $$ where $\xi_j$ are the eigenvectors, and $\lambda_j$ are the eigenvalues.
If the coefficients (or initial condition) are chosen so $c_1 = 1$ and $c_2 = 0$ then the solution is
$$ x(t) = \xi_1 e^{\lambda_1 t}. $$
If $\xi_1 = \begin{pmatrix}a \\ b\end{pmatrix}$, then this solution can be written $$ \begin{pmatrix}x_1(t) \\ x_2(t) \end{pmatrix} = \begin{pmatrix}a e^{\lambda_1 t} \\ b e^{\lambda_1 t}\end{pmatrix} $$
and in particular, for all $t$, $x_1$ and $x_2$ satisfy
$$ x_2 = b e^{\lambda_1 t} = b\left(\dfrac{1}{a} x_1\right) = \dfrac{b}{a}x_1 $$
i.e. the entire solution lies in the line passing through the origin, with slope $\dfrac{b}{a}$.