grandes-ecoles 2020 Q34

grandes-ecoles · France · centrale-maths1__mp Matrices Eigenvalue and Characteristic Polynomial Analysis

Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ and $v$ be two endomorphisms of $E$ such that, for all $k \in \mathbb{N}, \operatorname{Tr}\left(u^k\right) = \operatorname{Tr}\left(v^k\right)$. Show that $u$ and $v$ have the same characteristic polynomial.

Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ and $v$ be two endomorphisms of $E$ such that, for all $k \in \mathbb{N}, \operatorname{Tr}\left(u^k\right) = \operatorname{Tr}\left(v^k\right)$. Show that $u$ and $v$ have the same characteristic polynomial.

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