Let A be a Hermitian matrix.

Then, by definition:

where A∗ denotes the conjugate transpose of A.

Let λ be an eigenvalue of A.

Let v be an eigenvector corresponding to the eigenvalue λ.

By definition of eigenvector:

Left-multiplying both sides by v∗, we obtain:

Firstly, note that both v∗Av and v∗v are 1×1-matrices.

Now observe that, using Conjugate Transpose of Matrix Product: General Case:

As A is Hermitian, and (v∗)∗=v by Double Conjugate Transpose is Itself, it follows that:

That is, v∗Av is also Hermitian.

By Product with Conjugate Transpose Matrix is Hermitian, v∗v is Hermitian.

So both v∗Av and v∗v are Hermitian 1×1 matrices.

Now suppose that we have for some a,b∈C:

Note that b≠0 as an Definition:Eigenvector is non-zero.

By definition of a Hermitian matrix:

where a¯ denotes the complex conjugate of a.

By Complex Number equals Conjugate iff Wholly Real, it follows that a,b∈R, that is, are real.

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