= Symplectic group
{title2=$Sp(n, F)$}
Intuition, please? Example? https://mathoverflow.net/questions/278641/intuition-for-symplectic-groups The key motivation seems to be related to <Hamiltonian mechanics>. The two arguments of the <bilinear form> correspond to each set of variables in Hamiltonian mechanics: the generalized positions and generalized momentums, which appear in the same number each.
Seems to be set of matrices that preserve a <skew-symmetric bilinear form>, which is comparable to the <orthogonal group>, which preserves a <symmetric bilinear form>. More precisely, the orthogonal group has:
$$
O^T I O = I
$$
and its generalization the <indefinite orthogonal group> has:
$$
O^T S O = I
$$
where S is symmetric. So for the symplectic group we have matrices Y such as:
$$
Y^T A Y = I
$$
where A is antisymmetric. This is explained at: https://www.ucl.ac.uk/~ucahad0/7302_handout_13.pdf They also explain there that unlike as in the analogous <orthogonal group>, that definition ends up excluding determinant -1 automatically.
Therefore, just like the <special orthogonal group>, the symplectic group is also a <subgroup> of the <special linear group>.