For every even dimension , the symplectic group is the group of matrices which preserve a nondegenerate antisymmetric bilinear form , 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.
Since is a volume form, the symplectic group preserves volume and vector space orientation. Hence, . In fact, is just the group of matrices with determinant 1. The three symplectic (0,1)-matrices are therefore
(1)
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The matrices
(2)
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and
(3)
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are in , where
(4)
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In fact, both of these examples are 1-parameter subgroups.
A matrix can be tested to see if it is symplectic using the Wolfram Language code:
SymplecticForm[n_Integer] := Join[PadLeft[IdentityMatrix[n], {n, 2n}], PadRight[-IdentityMatrix[n], {n, 2n}]] SymplecticQ[a_List]:= EvenQ[Length[a]]&& Transpose[a] . SymplecticForm[Length[a]/2] . a == SymplecticForm[Length[a]/2]
Thinking of a matrix as given by coordinate functions, the set of matrices is identified with . The symplectic matrices are the solutions to the equations
(5)
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where is defined by
(6)
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Note that these equations are redundant, since only of these are independent, leaving "free" variables. In fact, the symplectic group is a smooth -dimensional submanifold of .
Because the symplectic group is a group and a manifold, it is a Lie group. Its submanifold tangent space at the identity is the symplectic Lie algebra . The symplectic group is not compact.
Instead of using real numbers for the coefficients, it is possible to use coefficients from any field . The symplectic group for even is the group of elements of the general linear group that preserve a given nonsingular symplectic form. Any such matrix has determinant 1.