Classify up to similarity all $n\times n$ complex matrices such that $A^n=I$

Question

Classify up to similarity all $n\times n$ complex matrices such that $A^n=I$.

I've seen this question for $n=3$. But I was wondering how to generalize this result.

First $A^n=I$ is diagonalizable. So it is similar to some diagonalizable matrix. So it's minimal polynomial is the product of distinct linear factors.

Let $J$ be the Jordan form of $A$. I believe that all the Jordan blocks should be size one. Since it is able to be factored into distinct monic polynomials. This means each root will have multiplicity of $1$. Hence each Jordan block will be size one. So $A$ is similar to a diagonal matrix with distinct eigenvalues.

Answer 1

It seems to me that the answers, especially khalatnikov's, extend to arbitrary n. Your matrix is, up to similarity, a diagonal matrix in which all diagonal entries are nth roots of unity.

So A is similar to a diagonal matrix with distinct eigenvalues.

The identity matrix is a trivial solution, but it does not have distinct eigenvalues.