Let λ be an arbitrary eigenvalue of a Hermitian matrix A and let x be an eigenvector corresponding to the eigenvalue λ.
Then we have
Multiplying by x¯T from the left, we obtain
We also have
The first equality follows because the dot product u⋅v of vectors u,v is commutative.
That is, we have
We applied this fact with u=x¯ and v=Ax.
Thus we obtain
Taking the complex conjugate of this equality, we have
(Note that x¯¯=x. Also ||x||¯¯¯¯¯¯¯¯¯=||x|| because ||x|| is a real number.)
Since the matrix A is Hermitian, we have A¯T=A.
This yields that
Recall that x is an eigenvector, hence x is not the zero vector and the length ||x||≠0.
Therefore, we divide by the length ||x|| and get
It follows from this that the eigenvalue λ is a real number.
Since λ is an arbitrary eigenvalue of A, we conclude that all the eigenvalues of the Hermitian matrix A are real numbers.