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symplectic matrix∗
matte†
2013-03-21 15:50:11
A real 2n × 2n matrix A ∈ M2n (R) is a symplectic matrix if AJAT = J,
where AT is the transpose of A, and J ∈ O(2n) is the orthogonal matrix
0
In
J=
.
−In 0
Here In ∈ Mn (R) is the identity n × n matrix and 0 ∈ Mn (R) is the zero n × n
matrix.
Symplectic matrices satisfy the following properties:
1. The determinant of a symplectic matrix equals one.
2. With standard matrix multiplication, symplectic 2n × 2n matrices form a
group denoted by Sp(2n).
A B
3. Suppose Ψ =
, where A, B, C, D are n × n matrices. Then Ψ is
C D
symplectic if and only if
ADT − BC T = I,
AB T = BAT ,
CDT = DC T .
4. If X and
Y are real
n × n matrices, then U = X + iY is unitary if and
X −Y
only if
is symplectic.
Y
X
∗ hSymplecticMatrixi
created: h2013-03-21i by: hmattei version: h34140i Privacy setting:
h1i hDefinitioni h53D05i
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