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

Statement

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 , i.e.,

.

Proof

It is given that

,
.

Since , it follows that

.

However, we have

.

Therefore,

.

Therefore, ,

and

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.