I was studying Jordan descomposition of a Matrix, and I am really stuck in the algorithm to do this.
I start with the matrix $$A=\begin{pmatrix}1 & -3 & 1\\ 1&5&-1\\ 2&6&0 \end{pmatrix}.$$
I obtain the characteristic polynomial which is $-(x-2)^3,$ and then I find the eigenvectors, but when i try to make the canonical form, i can't find the correct generalized eigenvalue, because $(A-2I)x=v,$ with either $v$ eigenvector, has no solution.
I get the next eigenvectors: $v_1=(0,1,3),v_2=(1,0,1).$ Computing it online, the matrix $S$ (where $A=S J S^{-1}$), is different. The first column is the first $v_1$ but the second column is $v2-v1$ and this is where i get very confused.
Thanks for any help!