Minimal polynomial and diagonalization of a block matrix

[mahler1]

Homework Statement .

Let $X:=\{A \in \mathbb C^{n\times n} : rank(A)=1\}$. 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 $n\times n$ 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.

 

[AlephZero]

I know that, thinking in terms of columns X is the set of matrices with just one linearly independent column.

That is correct, but rank-1 matrices have other properties that could be more useful to answer the question. Reading http://en.wikipedia.org/wiki/Matrix_similarity should give you some ideas.

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.

It's not true. There are an infinite number of equivalence classes, and an infinite number of matrices in each class.