As the title says, how do I classify up to similarity all nxn nilpotent matrices over K for n less than or equal to 5? I have no clue, any thoughts?
Maybe the case $\displaystyle n=2$ can give you ideas. Let $\displaystyle A$ be a $\displaystyle 2\times 2$ nilpotent matrix. Either $\displaystyle A=0$ or $\displaystyle A^2=0$. If $\displaystyle A\neq 0$, let $\displaystyle x$ be such that $\displaystyle Ax\neq 0$. Then $\displaystyle (x,Ax)$ forms a basis of $\displaystyle K^2$, and in this basis, $\displaystyle A$ looks like $\displaystyle \begin{pmatrix}0&1\\0&0\end{pmatrix}$.