Jordan canonical form of a matrix for distinct eigenvalues How can i find the Jordan canonical form of this matrix $$A=\begin{pmatrix}1 &0 &0 &0 \\ 1& 2& 0& 0\\ 1 &0& 2& 0\\ 1 &1& 0& 2\end{pmatrix}.$$
In my book there are examples but all the matrices in these examples have only one eigenvalue repeated n times(for $n\times n$ matrices) but the matrix $A$ has eigenvalues $1$ and $2$(multiple of $3$). What is the way of finding $A$'s jordan canonical form?
Thanks

The properties listed here can help you.
In this case you have three possibilities for the Jordan Canonical Form.These are:
$$J_1=\begin{pmatrix}1 &0 &0 &0 \\ 0& 2& 0& 0\\ 0 &0& 2& 0\\ 0 &0& 0& 2\end{pmatrix} , \ J_2=\begin{pmatrix}1 &0 &0 &0 \\ 0& 2& 0& 0\\ 0 &0& 2& 1\\ 0 &0& 0& 2\end{pmatrix}, \ J_3=\begin{pmatrix}1 &0 &0 &0 \\ 0& 2& 1& 0\\ 0 &0& 2& 1\\ 0 &0& 0& 2 \end{pmatrix}.$$
If $(A-I)(A-2I)=0$ is $J_1$. If $(A-I)(A-2I)\neq 0$ and $(A-I)(A-2I)^2= 0$ is $J_2$. Else is $J_3.$
