For every even dimension 2n, the symplectic group Sp(2n) is the group of 2n×2n matrices which preserve a nondegenerate antisymmetric bilinear form ω, i.e., a symplectic form. Every symplectic form can be put into a canonical form by finding a symplectic basis. So, up to conjugation, there is only one symplectic group, in contrast to the orthogonal group which preserves a nondegenerate symmetric bilinear form. As with the orthogonal group, the columns of a symplectic matrix form a symplectic basis.
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