In this post, we prove the following:
Statement: All eigenvalues of a Hermitian matrix are real.
Since the matrix is Hermitian, by definition,
Let be an eignevalue of , and be the corresponding eigenvector. Let
The sum in the right hand side is real. It follows that will be real if and only if is real. Let us examine: Using ,
thus must be real. Consequently, must be real.
In another post, we have shown that for a Hermitian matrix, the eigenvectors pertaining to different eigenvalues are orthogonal.