Solution
= Solution
Let $v\ne0$ be an <eigenvector> of the <Hermitian matrix> $A$, with <eigenvalue> $\lambda$. Multiplying $Av=\lambda v$ by $v^\dagger$ gives
$$
\lambda=\frac{v^\dagger Av}{v^\dagger v}.
$$
The denominator is the positive real number $\|v\|^2$, while
$$
\overline{v^\dagger Av}=v^\dagger A^\dagger v=v^\dagger Av.
$$
Thus the numerator is real, and \b[every <eigenvalue> of a <Hermitian matrix> is real]. This argument does not require distinct <eigenvalues>.