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

Statement

Let M be an n\times n complex Hermitian matrix which means M=M^* where * denote the conjugate transpose operation. Let \lambda_1, \neq \lambda_2 be two different  eigenvalues of M. Let x, y be the two eigenvectors of M corresponding to the two eigenvalues \lambda_1 and \lambda_2, respectively.

Then the following is true:

\boxed{\lambda_1 \neq \lambda_2 \iff \langle x, y \rangle = 0.}

Here \langle a,b\rangle denote the usual inner product of two vectors x,y, i.e.,

\langle x,y \rangle := y^*x.

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This is elementary, yet useful and interesting.

Statement: All eigenvalues of a Hermitian matrix are real.

Proof:

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