Recently I was studying my notes from my Lie algebra class and I came with this definition of the Symplectic group:
$$Sp(2n,\mathbb{K}):= \{A\in M(2n,\mathbb{K}):(Av,Aw)_{Sp}=(v,w)_{Sp}\} $$ With $(Av,Aw)_{Sp}:=\sum^{n}_{i=1}v^{i}w^{n+i}-\sum^{n}_{j=1}v^{n+j}w^{j}$
I found this definition really interesting as it was the first time that I saw it. With this, I saw the classic definition of the Symplectic group which is given by $Sp(2n,\mathbb{K}):= \{A\in M(2n,\mathbb{K}):A^{T}JA = J\}$ with $J=\begin{bmatrix}0 & -I \\\ I & 0\end{bmatrix}$.
So considering this definition this condition came up:
$A\in Sp(2n,\mathbb{k})$ if and only if $A^{T}JA = J$ with $J=\begin{bmatrix}0 & -I \\\ I & 0\end{bmatrix}$
How could this be proven? I have tried doing it, but I have not been able to achieve anything, any help would be really appreciated.
This is simple algebraic manipulation. One needs to show that for a matrix $A$, we have $$(Av, Aw)_{Sp} = (v, w)_{Sp}$$ for all $v, w$ if and only if $A^\top JA = J$, which follows easily from the fact that $$(v, w)_{Sp} = v^\top J w$$ for all $v, w$.