TITLE: Let $N$ be a $2× 2$ complex matrix such that $N^2=0$. how could I show $N=0$, or $N$ is similar over the matrix.
QUESTION [6 upvotes]: Let $N$ be a $2× 2$ complex matrix such that $N^2=0$. how could I show $N=0$, or $N$ is similar over $\mathbb{C}$ to \begin{bmatrix}0 & 0\\1 & 0\end{bmatrix}

REPLY [11 votes]: Let $J=\begin{bmatrix}0 & 0\\1 & 0\end{bmatrix}$. When $N\not=0$, there exists a nonzero vector $x$ such that $Nx\not=0$. Clearly $x$ and $Nx$ are linearly independent, otherwise $Nx=cx$ for some $c\neq0$ and in turn $0=N^2x=N(Nx)=cNx\neq0$, which is a contradiction. Therefore $P=(x,Nx)$ is invertible. Now you may verify that $NP=(Nx,N^2x)=(Nx,0)=PJ$. So, $N=PJP^{-1}$.