I would like to know the properties of orthogonal matrices and symplectic matrices in terms of the forms they preserve. Could someone please add and/or correct, maybe give some refs/examples?
AFAIK, given a quadratic form q on a vector space V over a field F, there is an associated orthogonal group O(2n) ,a subgroup of GL(n,F), which preserve q; if F is the reals O(2n) preserves q= inner-product and norm (since in R, the norm is induced by the inner-product). Symplectic matrices only preserve symplectic forms, i.e., bilinear,antisymmetric,non-degenerate forms.
Are there relations between these groups; do they overlap, intersect, etc?
I am interested mostly in the case where the field is Z/2.