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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.,

.

We will show that the eigenvalues of the Laplacian matrix of the complete graph are where the eigenvalue has (algebraic) multiplicity and the eigenvalue has multiplicity .

This is elementary, yet useful and interesting.

**Statement:** All eigenvalues of a Hermitian matrix are real.

**Proof:**