Problem from Hoffman-Kunze Chapter -6 I have started reading "Elementary Canonical Forms " from  Hoffman Kunze .
I am stuck on this:
Let $N$ be a $2\times 2$ complex matrix such that $N^2=0$ .Prove that either $N=0$ or $N=$ is similar to \begin{bmatrix} 0 & 0\\1 & 0 \end{bmatrix} 
My try;
Let $ N$=\begin{bmatrix} a & b\\ c & d\end{bmatrix}.On using that $N^2=0$ we get  conditions 
$a^2+bc=0;(a+d)b=0;(a+d)c=0;bc+d^2=0$. On solving we get either $a=d $ or $a=-d$.
If $a=d$ then $a=d=b=c=0 $ If $a=-d $ then nothing significant is obtained .How to show the second case?
 A: If $N^{2}=0$ but $N\ne 0$, then there is a vector $x$ such that $Nx \ne 0$ and $N^{2}x=0$. Then $\{ x, Nx \}$ is a basis of $\mathbb{R}^{2}$ because $\alpha x + \beta Nx = 0$ gives $N(\alpha x+\beta Nx) = \alpha Nx = 0$ and, hence, $\alpha =0$ (and $\beta=0$ then also follows easily.) Using this basis, look at $N$:
$$
                      x \mapsto Nx,\;\;\; Nx \mapsto 0.
$$
The matrix representation of $N$ in this basis is the stated matrix because it maps the first basis element to the second, and the second one to $0$.
A: Most likely it means that $N = \left(\begin{array} &0 & 0 \\
1 &0 
\end{array} \right)$
in some basis, which means that any nilpotent matrix must be conjugate to that element. Solve the equation $\left(\begin{array} &e & f \\
g &h 
\end{array} \right)\left(\begin{array} &a & b \\
c &d 
\end{array} \right)\left(\begin{array} &e & f \\
g &h 
\end{array} \right)^{-1} = \left(\begin{array} &0 & 0 \\
1 &0 
\end{array} \right)$
and you'll get equivalent conditions to the ones you posted
