I have this differential equation: $$y'=Ay \text{ with } A=\begin{pmatrix}3&2\\-2&-1\end{pmatrix}$$ I have calculated the eigenvalues and found $\lambda_{1/2}=1$
Now if i replace lambda with the eigenvalue in the following matrix i am finding an eigenvector:
$$\begin{pmatrix}3-1&2\\-2&-1-1\end{pmatrix}=\begin{pmatrix}2&2\\-2&-2\end{pmatrix}=\begin{pmatrix}2&2\\0&0\end{pmatrix} => y=1 , x=-y$$ Hence: Eigenvector $\upsilon_{1}=\begin{pmatrix}-1\\1\end{pmatrix}$
My Problem is, that i cannot find the second eigenvector. I need a second one for creating a fundamental system.
How can i do that?
If you just want to solve the system above, there are other ways that do not require generalized eigenvectors. For example, in your specific case above, letting $y=(y_1,y_2)$, adding the first and the second row of the matrix gives $$\dot{y}_1+\dot{y}_2=y_1+y_2,$$ which gives $y_1+y_2=e^t+c$, and then solve a linear differential equation to find $y_1$ and $y_2$.