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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
M T Ω M = Ω , {\displaystyle M^{\text{T}}\Omega M=\Omega ,}