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Eigenvalues of a hermitian matrix

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Let A <- C^nxn be hermitian.
(a) Prove that all eigenvalues of A are real.
(b) Prove that if x and y are eigenvectors associated to distinct eigenvalues, then they are orthogonal , i.e., x^H y = 0.

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Please see the attached proof.

Proof.
(a) As A is Hermitian, we have

i.e.,

So, we take transpose on both sides to get
............................(1)
Assume that A has an eigenvalue , and its corresponding eigenvector is . Then we have
...........................(2)
So, taking conjugate on both sides, we get
.................(3)
So, by ...

Solution Summary

Eigenvalues of a hermitian matrix are clearly determined.

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