Symplectic matrix

In mathematics, a symplectic matrix is a 2n×2n matrix M with real entries that satisfies the condition

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

 

 

 

 

()

where M^T denotes the transpose of M and Ω is a fixed 2n×2n nonsingular, skew-symmetric matrix. This definition can be extended to 2n×2n matrices with entries in other fields, e.g. the complex numbers.

Typically Ω is chosen to be the block matrix

where I_n is the n×n identity matrix. The matrix Ω has determinant +1 and has an inverse given by Ω⁻¹ = Ω^T = −Ω.

Every symplectic matrix has unit determinant, and the 2n×2n symplectic matrices with real entries form a subgroup of the special linear group SL(2n, R) under matrix multiplication, specifically a connected noncompact real Lie group of real dimension n(2n + 1), the symplectic group Sp(2n, R). The symplectic group can be defined as the set of linear transformations that preserve the symplectic form of a real symplectic vector space.

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