grandes-ecoles 2022 Q10

grandes-ecoles · France · mines-ponts-maths2__mp Matrices Matrix Group and Subgroup Structure

In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$
$\mathbf{10}$ ▷ If $G = \mathrm{O}_n(\mathbf{R})$, show that $\mathcal{A}_G = \mathcal{A}_n(\mathbf{R})$, the set of antisymmetric matrices.

In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra:
$$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$

$\mathbf{10}$ ▷ If $G = \mathrm{O}_n(\mathbf{R})$, show that $\mathcal{A}_G = \mathcal{A}_n(\mathbf{R})$, the set of antisymmetric matrices.

Paper Questions

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