= Solution
A <Hermitian matrix> satisfies $H^\dagger=H$, where $\dagger$ denotes the <conjugate transpose>. For a nonzero <eigenvector> $v$ with $Hv=\lambda v$, the scalar $v^\dagger Hv$ equals its own <complex conjugate>, and
$$
\lambda=\frac{v^\dagger Hv}{v^\dagger v}\in\mathbb R.
$$
Thus \b[every Hermitian <eigenvalue> is real].
A <unitary matrix> satisfies $U^\dagger U=UU^\dagger=I$. If $Uv=\mu v$, preservation of the squared <norm> gives
$$
v^\dagger v=(Uv)^\dagger(Uv)=|\mu|^2v^\dagger v.
$$
Since $v\neq0$, $\boxed{|\mu|=1}$. These statements concern complex <eigenvectors>, even when the <matrix> entries happen to be real.
Back to article page
