This was [asked](https://math.stackexchange.com/questions/3012437/eigenvectors-of-nth-root-of-identity-matrix) 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?