What can be said about the rank of a nilpotent matrix?

Question

Is there anything that can be said about the rank of a nilpotent $n \times n$ matrix? It certainly is less than $n$ but anything else?

Answer 1

The rank can take any value between $0$ and $n-1$. To prove it, consider a basis $\mathcal B = (b_1, \dots, b_n)$ and the nilpotent matrix defined by $M.b_i = b_{i-1}$ for $2 \le i \le k \le n$ and $M.b_i=0$ otherwise. The rank of $M$ is equal to $k-1$ and $k$ can be any value between $2$ and $n$.

Add to this the zero operator which has rank $0$.