TITLE: Orthogonal Matrices and Symplectic Matrices and Preservation of Forms
QUESTION [5 upvotes]: 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.
Thanks

REPLY [3 votes]: Over the reals the intersection of the orthogonal and symplectic groups in even dimension is isomorphic
to the unitary group in the half dimension. 
$ U(n) = O(2n, \mathbf{R}) \cap Sp(2n, \mathbf{R})$
This is the 2 out of 3 property expressing the compatibility the symplectic structure with the symmetric bilinear form of the orthogonal group.
The orthogonal group over $Z_2$ is a subgroup of the symplectic group because a symmetric bilinear form is also alternating (since $-1 = +1$).
The full symplectic group $Sp(2n, Z_2)$ can be realized from the action of the $2^n$ dimensional Clifford group on the bits of the binary
representation of the basis vectors (up to  a phase) as explained in Daniel Gottesmann's
 paper.