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Symplectic matrix : ウィキペディア英語版
Symplectic matrix
In mathematics, a symplectic matrix is a 2''n''×2''n'' matrix ''M'' with real entries that satisfies the condition
where ''MT'' denotes the transpose of ''M'' and Ω is a fixed 2''n''×2''n'' nonsingular, skew-symmetric matrix. This definition can be extended to 2''n''×2''n'' matrices with entries in other fields, e.g. the complex numbers.
Typically Ω is chosen to be the block matrix
:\Omega =
\begin
0 & I_n \\
-I_n & 0 \\
\end
where ''I''n is the ''n''×''n'' identity matrix. The matrix Ω has determinant +1 and has an inverse given by Ω−1 = ΩT = −Ω.
Every symplectic matrix has unit determinant, and the 2''n''×2''n'' symplectic matrices with real entries form a subgroup of the special linear group SL(2''n'', ''R'') under matrix multiplication, specifically a connected noncompact real Lie group of real dimension , the symplectic group Sp(2''n'', R). The symplectic group can be defined as the set of linear transformations that preserve the symplectic form of a real symplectic vector space.
An example of a group of symplectic matrices is the group of three symplectic 2x2-matrices consisting in the identity matrix, the upper triagonal matrix and the lower triangular matrix, each with entries 0 and 1.
==Properties==

Every symplectic matrix is invertible with the inverse matrix given by
:M^ = \Omega^ M^\text \Omega.
Furthermore, the product of two symplectic matrices is, again, a symplectic matrix. This gives the set of all symplectic matrices the structure of a group. There exists a natural manifold structure on this group which makes it into a (real or complex) Lie group called the symplectic group.
It follows easily from the definition that the determinant of any symplectic matrix is ±1. Actually, it turns out that the determinant is always +1. One way to see this is through the use of the Pfaffian and the identity
:\mbox(M^\text \Omega M) = \det(M)\mbox(\Omega).
Since M^\text \Omega M = \Omega and \mbox(\Omega) \neq 0 we have that det(''M'') = 1.
Suppose Ω is given in the standard form and let ''M'' be a 2''n''×2''n'' block matrix given by
:M = \beginA & B \\ C & D\end
where ''A'', ''B'', ''C'', ''D'' are ''n''×''n'' matrices. The condition for ''M'' to be symplectic is equivalent to the conditions
:A^\text D - C^\text B = I
:A^\text C = C^\text A
:D^\text B = B^\text D.
When ''n'' = 1 these conditions reduce to the single condition det(''M'') = 1. Thus a 2×2 matrix is symplectic iff it has unit determinant.
With Ω in standard form, the inverse of ''M'' is given by
:M^ = \Omega^ M^\text \Omega=\beginD^\text & -B^\text \\-C^\text & A^\text\end.
The group has dimension ''n''(2''n'' + 1). This can be seen by noting that the group condition implies that
:\Omega M^\text \Omega M = -I
this gives equations of the form
: -\delta_ = \sum_^n m_m_ - m_m_ - m_m_ + m_m_
where m_ is the i,j-th element of M. The sum is antisymmetric with respect to indices i,j, and since the left hand side is zero when i differs from j, this leaves n(2n-1) independent equations.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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