"The set of eigenvalues of a matrix A of m-involution (for which $A^m=I$ for an integer m>1) belongs to the set of m-th roots of unity."
How do I prove this?
It can be shown for m=2, the eigenvalues are $\lambda = \sqrt1 = \pm 1$
$$
Ax = \lambda x \Leftrightarrow A^{-1}A x= \lambda A^{-1} x \Leftrightarrow x = \lambda A x \Leftrightarrow x = \lambda^2 x \Leftrightarrow (1-\lambda^2)x = 0
$$
then $\lambda =\pm 1$
how do I prove the generalised version?
$Ax=\lambda x, x\neq 0$ implies $x=A^{m}x=\lambda^{m}x$ so $\lambda^{m}=1$.
[$Ax=\lambda x$. Apply $A$ on both sides to get $A^{2}x=\lambda^{2} x$, and so on. By induction $A^{n}x=\lambda^{n} x$ for all $n$].