grandes-ecoles 2020 Q3

grandes-ecoles · France · centrale-maths1__pc Matrices Bilinear and Symplectic Form Properties
In the case $n=1$: Let $M$ be an orthogonal matrix of size $2 \times 2$. We denote by $M_{1} = \binom{x_{1}}{x_{2}}$ and $M_{2} = \binom{y_{1}}{y_{2}}$ the two columns of $M$. Show the equivalence $$M \text{ is symplectic } \Longleftrightarrow M_{2} = -J_{1} M_{1}.$$ 
In the case $n=1$: Let $M$ be an orthogonal matrix of size $2 \times 2$. We denote by $M_{1} = \binom{x_{1}}{x_{2}}$ and $M_{2} = \binom{y_{1}}{y_{2}}$ the two columns of $M$. Show the equivalence
$$M \text{ is symplectic } \Longleftrightarrow M_{2} = -J_{1} M_{1}.$$
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