<s>[INST] Jordan form hard problem I'm trying to find a matrix $P$ such that $J=P^{-1}AP$, where $J$ is the Jordan Form of the matrix:
$$A=\begin{bmatrix}{1}&{1}&{1}&{1}&{1}&{1}&{1}&{1}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}&{1}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}&{-1}\\{0}&{1}&{1}&{0}&{0}&{0}&{0}&{1}\\{0}&{0}&{0}&{1}&{1}&{0}&{0}&{0}\\{0}&{1}&{1}&{1}&{1}&{1}&{0}&{1}\\{0}&{-1}&{-1}&{-1}&{-1}&{0}&{1}&{-1}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\end{bmatrix}$$
$p(x)=x(x-1)^{4}$ is the characteristic polynomial.  Then, the eigenvalues are $ (1, 1, 1, 1, 0 ,0 ,0 ,0)$
I'm trying to find the eigenvectors by this way $(A-I)z=w$ for the eigenvectors with eigenvalue $1$ and $Ax=y$ for the eigenvectors with eigenvalue $0$. But this is impossible.
Can somebody help me please?
Thanks for your help.
 [/INST] Here is aJordan form computed by maple
$$  \left[ \begin {array}{cccccccc} 1&1&0&0&0&0&0&0\\0
&1&0&0&0&0&0&0\\ 0&0&0&1&0&0&0&0
\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&1&1&0
&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0
&0&1\\ 0&0&0&0&0&0&0&0\end {array} \right] 
.$$
The eigenvalues are given by 
$$ [0,0,0,0,1,1,1,1] $$
and the corresponding eigenvectors are the columns of the matrix
$$ \left[ \begin {array}{cccccccc} 0&0&0&0&1&0&0&0\\ 0
&-1&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0
\\ -1&0&0&0&0&0&0&0\\1&0&0&0&0&0&0
&0\\ 0&0&0&0&0&-1&0&0\\ 0&0&0&0&0&
1&0&0\\ 0&0&0&0&0&0&0&0\end {array} \right] $$
Now, you should be able to construct $P$.
</s>