Let A be a complex matrix of dimension $n$. Suppose that exists a natural number $r$ such that $A^r=c I_n$ , where $I_n$ is the identity matrix and $c$ is a complex number. For what values of $c$, is $A$ diagonalizable? I could only prove that $c$ must be different from zero.
Sorry if I didn't write everything in the correct way, but i'm new on this site. Thank you for the help.
If $c\neq 0$ you have $P=X^r-c$ which is a cancelling split (in $\mathbb C$) polynom with simple roots for $A$, hence $A$ is diagonalizable.