I've some troubles understanding Jordan matrices and generalized eigenvectors, and have a question regarding an exercise. I want to find the Jordan matrix for matrix $A$.
$$A=\begin{bmatrix} -2 & -1 & 1 \\ 5 & -1 & 4 \\ 5 & 1 & 2 \end{bmatrix}$$
Eigenvalues: $$p(\lambda)=(\lambda-3)(\lambda+2)^2$$
Eigenvector corresponding to $\lambda_1=3$ is $\vec{v_1}=(0,1,1)$
Eigenvector corresponding to $\lambda_2=-2$ is $\vec{v_2}=(-2,2,2)$
From here on the course literature finds generalized eigenvectors from $\vec{v_1}$, to be able to write the Jordan matrix $J=V^{-1}AV$
What is the argument for choosing from $\vec{v_1}$ and not $\vec{v_2}$? What difference does it make when choosing from which vector we find our generalized eigenvectors from?