Matrices#
Real symmetric matrix#
Orthogonal matrix#
Symplectic matrix#
Definition 1 (Symplectic matrix)
A symplectic matrix is a \(2n\times 2n\) matrix \(M\) with real entries that satisfies the condition
\[M^T \Omega M = \Omega,\]
where \(M^T\) denotes the transpose of \(M\) and \(\Omega\) is a fixed \(2n\times 2n\) nonsingular, skew-symemtric matrix.
Typically \(\Omega\) chosen to be the block matrix
\[\begin{split}\Omega = 0, I_n \\ -I_n, 0\end{split}\]
Remark 1
If instead M is a 2n × 2n matrix with complex entries, the definition is not standard throughout the literature. Many authors adjust the definition above to
\[M^* \Omega M = \Omega\,.\]
where M* denotes the conjugate transpose of M. In this case, the determinant may not be 1, but will have absolute value 1. In the 2×2 case (n=1), M will be the product of a real symplectic matrix and a complex number of absolute value 1.
Unitary matrix#
\[
U^{-1} = U^\dagger
\]