In mathematics, a symplectic matrix is a 2 n × 2 n {\displaystyle 2n\times 2n} matrix M {\displaystyle M} with real entries that satisfies the condition
where M T {\displaystyle M^{\text{T}}} denotes the transpose of M {\displaystyle M} and Ω {\displaystyle \Omega } is a fixed 2 n × 2 n {\displaystyle 2n\times 2n} nonsingular, skew-symmetric matrix. This definition can be extended to 2 n × 2 n {\displaystyle 2n\times 2n} matrices with entries in other fields, such as the complex numbers, finite fields, p-adic numbers, and function fields. Typically Ω {\displaystyle \Omega } is chosen to be the block matrix
Ω = [ 0 I n − I n 0 ] , {\displaystyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}},}
where I n {\displaystyle I_{n}} is the n × n {\displaystyle n\times n} identity matrix. The matrix Ω {\displaystyle \Omega } has determinant + 1 {\displaystyle +1} and its inverse is Ω − 1 = Ω T = − Ω {\displaystyle \Omega ^{-1}=\Omega ^{\text{T}}=-\Omega } .
Properties
Generators for symplectic matrices Every symplectic matrix has determinant + 1 {\displaystyle +1} , and the 2 n × 2 n {\displaystyle 2n\times 2n} symplectic matrices with real entries form a subgroup of the general linear group G L ( 2 n ; R ) {\displaystyle \mathrm {GL} (2n;\mathbb {R} )} under matrix multiplication since being symplectic is a property stable under matrix multiplication. Topologically, this symplectic group is a connected noncompact real Lie group of real dimension n ( 2 n + 1 ) {\displaystyle n(2n+1)} , and is denoted S p ( 2 n ; R ) {\displaystyle \mathrm {Sp} (2n;\mathbb {R} )} . The symplectic group can be defined as the set of linear transformations that preserve the symplectic form of a real symplectic vector space. This symplectic group has a distinguished set of generators, which can be used to find all possible symplectic matrices. This includes the following sets
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