# Nilpotent Matrices

## Homework Statement

Suppose that N is a nilpotent mxm matrix, N$^{m}$=0, but N$^{m'}$$\neq$0 for m'<m. Show that there exists a basis in which it takes the form of a single Jordan block with vanishing diagonal elements. Prove that your basis set is linearly independent.

## The Attempt at a Solution

So I've recognized the fact that N$^{m'}$$\neq$0 for m'<m means that N$^{m'}$ does not annihilate every vector in V. I'm just not really sure where go from here...