The problem statement, all variables and given/known data.
Let $X:=\{A ∈ \mathbb C^{n×n}:rank(A)=1\}$. Determine a representative for each equivalence class, for the equivalence relation "similarity" in $X$.
The attempt at a solution.
I am a pretty lost with this problem: I know that, thinking in terms of columns $X$ is the set of matrices with just one linearly independent column. In an $n×n$ matrix there are n columns, so I thought that maybe there could be n representatives of this equivalence relation, but I couldn't prove it and in fact I am not at all convinced this is true. I would appreciate suggestions to solve the problem.
Hint: If rank$(A)=1$, then any nonzero column of $A$ is an eigenvector of $A$, and two such matrices will be similar iff they have the same eigenvalue for their selected eigenvector.
(Note that eigenvalue $0$ is ruled out.)