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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 want to show that for , the following holds:

First, recall the power series . For , this sum is definitely larger than its th term only. In other words, , which implies . Now,

This proof is extracted from the StackExchange discussion here. For more inequalities/bounds for binomial coefficients, see Wikipedia.

This is elementary, yet useful and interesting.

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

**Proof:**