This is an elementary (yet important) fact in matrix analysis.
Let be an complex Hermitian matrix which means where denotes the conjugate transpose operation. Let be two different eigenvalues of . Let be the two eigenvectors of corresponding to the two eigenvalues and , respectively.
Then the following is true:
Here denotes the usual inner product of two vectors .
It is given that
Since , it follows that
However, we have
since must be real. Therefore,
Thus the eigenvectors corresponding to different eigenvalues of a Hermitian matrix are orthogonal. Additionally, the eigenvalues corresponding to a pair of non-orthogonal eigenvectors are equal.