symplectic matrix

A real 2n×2n matrix AM2n() is a symplectic matrix if AJAT=J, where AT is the transposeMathworldPlanetmath of A, and JO(2n) is the orthogonal matrixMathworldPlanetmath


Here 𝐈nMn() is the identity n×n matrix and 𝟎Mn() is the zero n×n matrix.

Symplectic matrices satisfy the following properties:

  1. 1.

    The determinantMathworldPlanetmath of a symplectic matrix equals one.

  2. 2.

    With standard matrix multiplication, symplectic 2n×2n matrices form a group denoted by Sp(2n).

  3. 3.

    Suppose Ψ=(ABCD), where A,B,C,D are n×n matrices. Then Ψ is symplectic if and only if

  4. 4.

    If X and Y are real n×n matrices, then U=X+iY is unitary if and only if (X-YYX) is symplectic.

Title symplectic matrix
Canonical name SymplecticMatrix
Date of creation 2013-03-22 13:32:28
Last modified on 2013-03-22 13:32:28
Owner matte (1858)
Last modified by matte (1858)
Numerical id 11
Author matte (1858)
Entry type Definition
Classification msc 53D05