I have this matrix $A=\left[ {\begin{array}{ccc} -3&3&-2\\ -7&6&-3\\ 1&-1&2\\ \end{array} } \right]$
I computed the characteristic polynomial $C_A(t)=-(t-2)^2(t-1)$
When I go to try and find the eigenvectors and generalized eigenvectors I compute $A-2I=\left[ {\begin{array}{ccc} -5&3&-2\\ -7&4&-3\\ 1&-1&0\\ \end{array} } \right]$
Using row reduction $\left[ {\begin{array}{ccc} 1&-1&0\\ 0&1&1\\ 0&0&0\\ \end{array} } \right]x=0$ gives me eigenvectors having the form $e=r\left[ {\begin{array}{c} 1\\ 1\\ -1\\ \end{array} } \right]$
$v=\left[ {\begin{array}{c} 1\\ 1\\ -1\\ \end{array} } \right]$ Is the only eigenvector up to a constant. So I know there is 1 generalized eigenvector and I simply chose this eigenvector to be my initial vector in the cycle.
At this point I know the Jordan form is $J=\left[ {\begin{array}{ccc} 2&1&0\\ 0&2&0\\ 0&0&1\\ \end{array} } \right]$
Now to find the generalized eigenvector I can take an augmented matrix $\left(\begin{array}{ccc|c} 5&3&-2&1\\ -7&4&1&1\\ 1&-1&0&-1 \end{array}\right)$ I row reduced this to the form $\left(\begin{array}{ccc|c} 1 & -1 & 0&-1\\ 0 & 1 & 1&2\\ 0&0&0&0 \end{array}\right)$
So I need a vector that satisfies $x_1-x_2=-1$ and $x_2+x_3=2$ I let $x_3=1$
Which gives me a vector $x=\left(\begin{array}{c} 0\\ 1\\ 1 \end{array}\right)$ which seems to work, so I have a basis for the generalized eigenspace for $\lambda=2$
then I need to find a solution for $(A-I)x=0$ for $\lambda=1$.
I row reduce $A-I=\left[\begin{array}{ccc} -4&3&-2\\ -7&5&1\\ 1&-1&1 \end{array}\right]$ to get $\left[\begin{array}{ccc} 1&0&-1\\ 0&1&-2\\ 0&0&0 \end{array}\right]$ and using $(A-I)x=0$ found that all eigenvectors for $\lambda=1$ have the form $v=s\left(\begin{array}{c} 1\\ 2\\ 1 \end{array}\right)$
So I can make a matrix $Q=\left[\begin{array}{ccc} 1&1&0\\ 2&1&1\\ 1&-1&1 \end{array}\right]$ s.t $A=QJQ^{-1}$ where the columns of $Q$ are a jordan basis for $J$
Because you are looking for a Jordan canonical basis, $Ax = 2x+v$, where $x$ is the generalized eigenvector you're looking for. Equivalently, $(A-2I)x = v$. Applying $A-2I$ one more time to each side will give you the following statement: $(A-2I)^{2}x = 0$.
So, your generalized eigenvector is in the nullspace of $(A-2I)^2$, and it is not in the nullspace of $(A-2I)$. Can you finish from there?