All nilpotent $2\times 2$ matrices 
I want to find all nilpotent $2\times 2$ matrices.

All nilpotent $2 \times 2$ matrices are similar($A=P^{-1}JP$) to $J = \begin{bmatrix} 0&1\\0&0\end{bmatrix}$
But how do I find all of these matrices?
I do think that the only such cases are $J$ and $J^
T$
 A: Suppose that $A$ ($2\times 2$) is nilpotent. Then $\det(A)$ is $0$, implying that an eigenvalue of $A$ is real and $0$. Because $A$ is $2\times 2$, there is one other eigenvalue which must also be $0$. (If $A^n=0$, then the eigenvalues of $A^n$ are $0$ but these are just the eigenvalues of $A$ raised to the same power $n$.) At this point, you can infer that $\text{tr}(A)=0$ and $A$ itself must necessarily take the form
$$
A=\begin{pmatrix}x & a \\ b & -x\end{pmatrix},\quad x\in\mathbb{R},\quad ab=-x^2. \tag{$*$}
$$
It turns out that ($*$) is also sufficient for nilpotency:
$$
\begin{pmatrix}x & a \\ b & -x\end{pmatrix}\cdot \begin{pmatrix}x & a \\ b & -x\end{pmatrix}=\begin{pmatrix}x^2+ab & xa-ax \\ bx-xb & ab+x^2\end{pmatrix}=\begin{pmatrix}0 & 0 \\ 0 & 0\end{pmatrix}\cdot
$$
Thus $A$ is nilpotent iff $A$ takes the form given in ($*$).
A: My mistake... in my comment above, I should not have said to solve the equations. 
Rather: if you pick an arbitrary $P$, and compute $P^{-1} J P$, you get a nilpotent matrix. Using
$$
P = \begin{bmatrix} a & b \\ c & d\end{bmatrix}
$$
but writing $P J P^{-1}$, I got
$$
PJP^{-1} = D \cdot \begin{bmatrix} -ac & a^2 \\ -c^2 & ac\end{bmatrix}
$$
where $D$ is the determinant $(ad - bc)$. 
Such matrices (for $a \ne 0$) have the general form 
$$
A = \begin{bmatrix} -S & T \\ -\frac{S^2}{T} & S\end{bmatrix}.
$$
and this is a "parameterization" (with parameters $S$ and $T$) of almost all possible nilpotent matrices. We also need to add in the $a = 0$ case, i.e.
$$
A = \begin{bmatrix} 0 & 0 \\ c & 0\end{bmatrix}.
$$
Note that in this parameterization, it's essential that $T \ne 0$. Because of this, you can say that up to scalar multiples, all nilpotent matrices have the form 
$$
A = \begin{bmatrix} -S & 1 \\ -S^2 & S\end{bmatrix}.
$$
or
$$
A = \begin{bmatrix} 0 & 0 \\ 1 & 0\end{bmatrix}.
$$
