grandes-ecoles 2020 Q20

grandes-ecoles · France · centrale-maths1__pc Matrices Eigenvalue and Characteristic Polynomial Analysis

Let $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Let $\lambda \in \mathrm{sp}_{\mathbb{R}}(M)$ and $p = \dim E_{\lambda}$. Let $(X_{1}, \ldots, X_{p})$ be a basis of $E_{\lambda}$. Show that $(J_{n} X_{1}, \ldots, J_{n} X_{p})$ is a basis of $E_{1/\lambda}$ and that $$\dim(E_{\lambda}) = \dim(E_{1/\lambda}).$$

Let $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Let $\lambda \in \mathrm{sp}_{\mathbb{R}}(M)$ and $p = \dim E_{\lambda}$. Let $(X_{1}, \ldots, X_{p})$ be a basis of $E_{\lambda}$. Show that $(J_{n} X_{1}, \ldots, J_{n} X_{p})$ is a basis of $E_{1/\lambda}$ and that
$$\dim(E_{\lambda}) = \dim(E_{1/\lambda}).$$

Paper Questions

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