= Solution
For the <unitary matrix> $C$, $C^{-1}=C^\dagger$. Therefore $D=C^{-1}AC$ is a <Hermitian matrix>:
$$
D^\dagger=(C^\dagger AC)^\dagger=C^\dagger AC=D.
$$
Every positive integer power remains a <Hermitian matrix>, since $(D^p)^\dagger=(D^\dagger)^p=D^p$. Its <eigenvalues> are real by the <finite-dimensional spectral theorem>. Equivalently, <matrix similarity> gives
$$
D^p=C^{-1}A^pC,
$$
whose <eigenvalues> are $\boxed{\lambda_j^p\in\mathbb R}$, where the $\lambda_j$ are the real <eigenvalues> of $A$.
Back to article page
