# Math Help - Classify up to similarity all nxn nilpotent matrices over K for n<=5

1. ## Classify up to similarity all nxn nilpotent matrices over K for n<=5

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?

2. ## Re: Classify up to similarity all nxn nilpotent matrices over K for n<=5

Maybe the case $n=2$ can give you ideas. Let $A$ be a $2\times 2$ nilpotent matrix. Either $A=0$ or $A^2=0$. If $A\neq 0$, let $x$ be such that $Ax\neq 0$. Then $(x,Ax)$ forms a basis of $K^2$, and in this basis, $A$ looks like $\begin{pmatrix}0&1\\0&0\end{pmatrix}$.