Graduate Are Eigenvalues of Hermitian Integer Matrices Always Integers?

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Eigenvalues of Hermitian matrices with integer entries are not guaranteed to be integers. The determinant equation, det(A - λI) = 0, indicates that polynomials with integer coefficients can have non-integer roots. This raises the possibility that while the matrices are Hermitian, their eigenvalues may not necessarily be integers. A counter-example is suggested as a potential way to explore this further. The discussion emphasizes the need for more investigation into the properties of such matrices and their eigenvalues.
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If matrix has integer entries and it is hermitian, are then eigenvalues also integers? Is there some theorem for this, or some counter example?
 
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LagrangeEuler said:
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?
 
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. :)
 
LagrangeEuler said:
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.
 

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