grandes-ecoles 2018 Q24

grandes-ecoles · France · centrale-maths1__psi Roots of polynomials Eigenvalue-root connection for matrices or linear operators

Let $(a_0, \ldots, a_{n-1}) \in \mathbb{C}^n$. Let $\lambda$ be a complex number. By discussing in $\mathbb{C}^n$ the system $C(a_0, \ldots, a_{n-1})X = \lambda X$, show that $\lambda$ is an eigenvalue of $C(a_0, \ldots, a_{n-1})$ if and only if $\lambda$ is a root of a polynomial of $\mathbb{C}[X]$ to be specified.

Let $(a_0, \ldots, a_{n-1}) \in \mathbb{C}^n$. Let $\lambda$ be a complex number. By discussing in $\mathbb{C}^n$ the system $C(a_0, \ldots, a_{n-1})X = \lambda X$, show that $\lambda$ is an eigenvalue of $C(a_0, \ldots, a_{n-1})$ if and only if $\lambda$ is a root of a polynomial of $\mathbb{C}[X]$ to be specified.

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