Find the Jordan normal form $J$ of the matrix

$$M=\left(\begin{array}{rrrr} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & -1 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right)$$

and determine both the characteristic and the minimal polynomial of $M$.

Find a basis of $\mathbb{C}^{4}$ such that $J$ (the Jordan normal form of $M$ ) is the matrix representing the endomorphism $M: \mathbb{C}^{4} \rightarrow \mathbb{C}^{4}$ in this basis. Give a change of basis matrix $P$ such that $P^{-1} M P=J$.