grandes-ecoles 2020 Q28

grandes-ecoles · France · centrale-maths1__pc Matrices Diagonalizability and Similarity

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Show that there exists $P \in \mathcal{O}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$ such that $P^{\top} M^{2} P$ is diagonal with diagonal coefficients $d_{1}, \ldots, d_{2n}$ satisfying for all $k \in \{1, \ldots, n\}$, $d_{k+n} = 1/d_{k}$.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Show that there exists $P \in \mathcal{O}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$ such that $P^{\top} M^{2} P$ is diagonal with diagonal coefficients $d_{1}, \ldots, d_{2n}$ satisfying for all $k \in \{1, \ldots, n\}$, $d_{k+n} = 1/d_{k}$.

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