In a Hermitian Matrix, the Eigenvectors of Different Eigenvalues are Orthogonal

This is an elementary (yet important) fact in matrix analysis.

Statement

Let be an complex Hermitian matrix which means where denote 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 denote the usual inner product of two vectors , i.e.,

.