Symplectic Matrix

Let M be a 2n×2n matrix with real entries. Then M is called a symplectic matrix if it satisfies the condition

(1)

where MT denotes the transpose of M and Ω is a fixed nonsingular, skew-symmetric matrix. Typically Ω is chosen to be the block matrix

\Omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix}

where In is the n×n identity matrix. Note that Ω has determinant +1 and has an inverse given by Ω−1 = ΩT = −Ω.

Read more about Symplectic Matrix:  Properties, Symplectic Transformations, The Matrix Ω, Complex Matrices

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