Linear Algebra - Elementary Canonical Forms Let $N$ be a $2\times2$ complex matrix such that $N^2=0$. Prove that either $N=0$ or $N$ is similar over $\mathbb{C}$ to 
$$\begin{pmatrix}0&0\\1&0\end{pmatrix}$$
 A: Hints:
$$N^2=0\iff \;\text{the characteristic polynomial of}\;N\;\;p_N(x)=x^2\implies\;\text{the minimal polynomial is either}\;\;x^2\;\;\text{or}\;\;x\ldots$$
Added: Trying to avoid working with the minimal polynomial: if $\,N^2=0\,$ then the only eigenvalue of $\,N\,$ is zero, so we have two possibilities: either there are two linearly independent eigevectors corresponding to this unique eigenvalue and then obviosuly $\,N=0\,$ , or else there is only one and then we can complete this eigenvector to a basis of $\,\Bbb C^2\,$ and get a matrix represention as the second option.
A: Hints: prove the following:
1) If $N^2=0$ then $\operatorname{Im}(N)\subseteq\ker(N)$.
2) Moreover, if $N\neq0$ and $N$ is $2\times2$, then $\operatorname{Im}(N)=\ker(N)$.
3) Take any $0\neq v\in\operatorname{Im}(N)=\ker(N)$. There exists $w\neq 0$ such that $Nw=v$ and $B=(w,v)$ form a basis to $\mathbb{C}^2$.
4) Denote $P=[I]^B_e$ - the identity matrix from basis $B$ to the standart basis $e$. What is $P^{-1}NP$?
