Are Eigenvalues of Hermitian Integer Matrices Always Integers?

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Discussion Overview

The discussion revolves around whether the eigenvalues of Hermitian matrices with integer entries are always integers. Participants explore the implications of this question, considering theorems and potential counterexamples.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants question if eigenvalues of Hermitian matrices with integer entries must be integers, suggesting there may be a theorem or counterexample related to this.
  • One participant notes that the characteristic polynomial derived from the determinant equation ##det(A-\lambda I)=0## has integer coefficients but does not necessarily have integer roots, implying that the eigenvalues may not be integers.
  • Another participant reiterates the point about the characteristic polynomial and expresses uncertainty about the implications of Hermitian properties on the eigenvalues.
  • There is a suggestion to look for a simple counterexample to clarify the situation.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are competing views regarding the nature of the eigenvalues of Hermitian integer matrices, with some expressing uncertainty and others proposing the need for counterexamples.

Contextual Notes

Participants acknowledge that the relationship between integer coefficients of the polynomial and the nature of its roots is crucial, but the implications of Hermitian properties remain unresolved.

LagrangeEuler
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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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