Are Eigenvalues of Hermitian Integer Matrices Always Integers?

[LagrangeEuler]

If matrix has integer entries and it is hermitian, are then eigenvalues also integers? Is there some theorem for this, or some counter example?

 

[PeroK]

If matrix has integer entries and it is hermitian, are then eigenvalues also integers? Is there some theorem for this, or some counter example?

What do you think? How do you find eigenvalues of a matrix?

 

[LagrangeEuler]

From $det(A-\lambda I)=0$. Polynomial with integer coefficients does not need to have integer roots. So I suppose that this is not the case. But here matrices are hermitian so I am not sure. :)

 

[PeroK]

From $det(A-\lambda I)=0$. Polynomial with integer coefficients does not need to have integer roots. So I suppose that this is not the case. But here matrices are symmetric so I am not sure. :)

I would think that would be the motivation to look for a simple counter-example.