grandes-ecoles 2020 Q1

grandes-ecoles · France · centrale-maths1__pc Matrices Bilinear and Symplectic Form Properties
In this question only, $n$ is any non-zero natural integer. Determine $J_{n}^{2}$ and show that $J_{n} \in \mathrm{Sp}_{2n}(\mathbb{R}) \cap \mathcal{A}_{2n}(\mathbb{R})$.
Recall: $J_{n} = \left(\begin{array}{cc} 0_{n,n} & I_{n} \\ -I_{n} & 0_{n,n} \end{array}\right)$, and a matrix $M \in \mathcal{M}_{2n}(\mathbb{R})$ is symplectic if and only if $M^{\top} J_{n} M = J_{n}$. 
In this question only, $n$ is any non-zero natural integer. Determine $J_{n}^{2}$ and show that $J_{n} \in \mathrm{Sp}_{2n}(\mathbb{R}) \cap \mathcal{A}_{2n}(\mathbb{R})$.

Recall: $J_{n} = \left(\begin{array}{cc} 0_{n,n} & I_{n} \\ -I_{n} & 0_{n,n} \end{array}\right)$, and a matrix $M \in \mathcal{M}_{2n}(\mathbb{R})$ is symplectic if and only if $M^{\top} J_{n} M = J_{n}$.
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