In mathematics, a symplectic matrix is a ${\displaystyle 2n\times 2n}$ matrix ${\displaystyle M}$ with real entries that satisfies the condition

where ${\displaystyle M^{\text{T}}}$ denotes the transpose of ${\displaystyle M}$ and ${\displaystyle \Omega }$ is a fixed ${\displaystyle 2n\times 2n}$ nonsingular, skew-symmetric matrix. This definition can be extended to ${\displaystyle 2n\times 2n}$ matrices with entries in other fields, such as the complex numbers, finite fields, p-adic numbers, and function fields.

Typically ${\displaystyle \Omega }$ is chosen to be the block matrix