What do the eigenvectors of the $n$th roots of $I_n$ look like? This was asked at math stackexchange a long time ago with no answers but some upvotes.
Let $A^n=I,$ where $A$ is $n\times n,$ and assume that $A^k\neq I,$ for all $1\leq k<n.$ Since its characteristic polynomial is $x^n-1$, the distinct $n^{th}$ roots of unity are its eigenvalues, 
Thus there are a full set of linearly independent eigenvectors.


*

*What do they look like?

*If we assume $A$ is orthogonal, what do they look like?

*If we assume $A$ is real, can one say anything more?

 A: Over $\mathbb C$, this seems pretty easy. Just put in Jordan normal form, which is necessarily diagonal (since $A^n=I$) with eigenvalues that are $n$th roots of unity (since $A^n=I$), and the lcm of the orders of the diagonal entries equals $n$ (since $A^k\ne I$ for $1\le k<n$). Then the eigenvectors are the standard basis vectors if the diagonal entries are distinct, but there will be higher dimensional eigenspaces if there are repeated eigenvalues.
A: Since $A^n=I$, $A$ is diagonalizable and eigenvalues are $n$-th roots of unity. They don't have to be all $n$-th roots of unity and they don't have to be distinct.
Your condition that $A^k\neq I$ simply means that for all $k<n$ not all
$\lambda_j^k$ are equal to $1$.
Speaking on eigenvectors, there are no restrictions: let $D$ be a diagonal matrix
whose elements are $n$-th roots of unity, and $B$ an arbitrary non-singular matrix.
Then $A=BDB^{-1}$ has the property that $A^n=I$ and eigenvectors are columns of $B$.
If $A$ is real, we have an additional property that eigenvalues come in conjugate pairs, and to conjugate eigenvalues correspond conjugate eigenvectors. Any $D$ and $B$ with these properties will give you $A$ with the required properties.
If $A$ is real and orthogonal, then we have the additional property that eigenvectors can be chosen to form an orthonormal basis. This means that $B$ above can be chosen unitary, so $A$ us unitary, and as it is real, it is orthogonal.
This is a complete description of eigenvalues and eigenvectors.
