grandes-ecoles 2020 Q10

grandes-ecoles · France · centrale-maths1__pc Matrices Bilinear and Symplectic Form Properties

With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Show that for all $(i,j) \in \{1,\ldots,2n\}^{2}$, $$\varphi(e_{i}, e_{j}) = \delta_{i+n,j} - \delta_{i,j+n}$$ (one may start with the case where $(i,j) \in \{1,\ldots,n\}^{2}$ then generalize).

With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Show that for all $(i,j) \in \{1,\ldots,2n\}^{2}$,
$$\varphi(e_{i}, e_{j}) = \delta_{i+n,j} - \delta_{i,j+n}$$
(one may start with the case where $(i,j) \in \{1,\ldots,n\}^{2}$ then generalize).

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