Correct aproach to find eigenvectors via eigenvalues Let $F=        \begin{bmatrix}
        -1 & -2 & 2 \\
        0 & 1 & 0 \\
        0 & 0 & 1 \\
        \end{bmatrix}$ be a matrix of a linear transformation $f:\mathbb{R^3}\rightarrow \mathbb{R^3}$.
Find the eigenvectors and eigenvalues for $F$.
Solving the $det(F-\lambda I)=0$ I get that $\lambda_{1}=\lambda_{2}=1,\lambda_{3}=-1$
For $\lambda_{1}=1$
the matrix $F-I$ looks like this
\begin{bmatrix}-2 & -2 & 2 \\0 & 0 & 0 \ \\0 & 0 & 0 \end{bmatrix}
The equivalent system for this matrix would be 
$-2x - 2y + 2z = 0$
Because $-2$ is the pivot which goes to $x$ I represent $x$ via $y,z$ so that $N(F-\lambda I)= L=(\begin{bmatrix}-1 \\1 \\0\end{bmatrix},\begin{bmatrix} 1 \\0 \\1\end{bmatrix})$ ?
Is that aproach correct?
I always represent the variables which multiply with the pivot via others?
 A: Yes (and this has nothing in particular to do with eigenvalues) when solving a linear system brought into row echelon form, the columns with a pivot correspond to the variables whose values are going to be determined by that equation; the columns without pivot are the variables whose values are chosen freely before this, and which serve as parameters for the solution. In your case you did not do much to get the system into row echelon form, so you are spoiled for choice of the pivot; however if you decide it is the initial term $-2x$ in the equation that is the pivot, then $y,z$ are parameters of the solution, and $x$ is determined by the equation. To get a basis of solutions, fix the parameters respectively to each column of the identity matrix, here $y=1,z=0$ respectively $y=0,z=1$, as you have done, giving the indicated vectors.
It should be noted that having $2$ independent eigenvectors for a double eigenvalue is quite exceptional; it would have been much more likely to find a system of rank$~2$ with only one independent solution. But people who make these exercises tend to have a strong preference for the diagonalisable case, which causes such extremely unlikely cases to often pop up in practical exercices.
