$A$ is a $n\times n$ matrix with elements in field $\textbf{k}$, such that $A^{T} = A^2$. Find all possible eigenvalues of $A$.
It is pretty easy to get $A^4=A$ from $A^{T}=A^2$, and then eigenvalue $\lambda$ must satisfy $\lambda=0,\;\lambda=1$ or $\lambda^3=1$. Examples for first two can be found easily, however I am not sure if every $\lambda^3=1$ can be eigenvalue of some matrix, could you help?