grandes-ecoles 2020 Q10

grandes-ecoles · France · x-ens-maths-a__mp_cpge Roots of polynomials Eigenvalue-root connection for matrices or linear operators

We say that a complex number $z$ is totally real (resp. totally positive) if there exists a non-zero polynomial $P ( X )$ with rational coefficients such that: (i) $z$ is a root of $P$, and (ii) all roots of $P$ are in $\mathbb { R }$ (resp. in $\mathbb { R } _ { + }$).
Let $M$ be a symmetric matrix with coefficients in $\mathbb { Q }$. Show that the eigenvalues of $M$ are totally real.

We say that a complex number $z$ is totally real (resp. totally positive) if there exists a non-zero polynomial $P ( X )$ with rational coefficients such that:
(i) $z$ is a root of $P$, and
(ii) all roots of $P$ are in $\mathbb { R }$ (resp. in $\mathbb { R } _ { + }$).

Let $M$ be a symmetric matrix with coefficients in $\mathbb { Q }$. Show that the eigenvalues of $M$ are totally real.

Paper Questions

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