Symplectic matrix

In mathematics, a symplectic matrix is a "2n"×"2n" matrix "M" (whose entries are typically either real or complex) satisfying the condition:$$M^T Omega M = Omega,.where "M^T" denotes the transpose of "M" and Ω is a fixed nonsingular, skew-symmetric matrix. Typically Ω is chosen to be the block matrix:$$Omega =egin{bmatrix}0 & I_n \-I_n & 0 \end{bmatrix}where "I"_n is the "n"×"n" identity matrix. Note that Ω has determinant +1 and has an inverse given by Ω⁻¹ = Ω^"T" = −Ω.

Properties

Every symplectic matrix is invertible with the inverse matrix given by:$$M^{-1} = Omega^{-1} M^T 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. The symplectic group has dimension "n"(2"n" + 1).

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:$$mbox{Pf}(M^T Omega M) = det(M)mbox{Pf}(Omega).Since $$M^T Omega M = Omega and $$mbox{Pf}(Omega) eq 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:$$M = egin{pmatrix}A & B \ C & Dend{pmatrix}where "A, B, C, D" are "n"×"n" matrices. The condition for "M" to be symplectic is equivalent to the conditions:$$A^TD - C^TB = I:$$A^TC = C^TA:$$D^TB = B^TD.

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.

ymplectic transformations

In the abstract formulation of linear algebra, matrices are replaced with linear transformations of finite-dimensional vector spaces. The abstract analog of a symplectic matrix is a symplectic transformation of a symplectic vector space. Briefly, a symplectic vector space is a 2"n"-dimensional vector space "V" equipped with a nondegenerate, skew-symmetric bilinear form ω called the symplectic form.

A symplectic transformation is then a linear transformation "L" : "V" → "V" which preserves ω, i.e.:$$omega(Lu, Lv) = omega(u, v).Fixing a basis for "V", ω can be written as a matrix Ω and "L" as a matrix "M". The condition that "L" be a symplectic transformation is precisely the condition that "M" be a symplectic matrix::$$M^T Omega M = Omega.

Under a change of basis, represented by a matrix "A", we have:$$Omega mapsto A^T Omega A:$$M mapsto A^{-1} M A.One can always bring Ω to either of the standard forms given in the introduction by a suitable choice of "A".

The matrix Ω

Symplectic matrices are defined relative to a fixed nonsingular, skew-symmetric matrix Ω. As explained in the previous section, Ω can be thought of as the coordinate representation of a nondegenerate skew-symmetric bilinear form. It is a basic result in linear algebra that any two such matrices differ from each other by a change of basis.

The most common alternative to the standard Ω given above is the block diagonal form:$$Omega = egin{bmatrix}egin{matrix}0 & 1\ -1 & 0end{matrix} & & 0 \ & ddots & \0 & & egin{matrix}0 & 1 \ -1 & 0end{matrix}end{bmatrix}.This choice differs from the previous one by a permutation of basis vectors.

Sometimes the notation "J" is used instead of Ω for the skew-symmetric matrix. This is a particularly unfortunate choice as it leads to confusion with the notion of a complex structure, which often has the same coordinate expression as Ω but represents a very different structure. A complex structure "J" is the coordinate representation of a linear transformation that squares to −1, whereas Ω is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which "J" is not skew-symmetric or Ω does not square to −1.

Given a hermitian structure on a vector space, "J" and Ω are related via:$$Omega_{ab} = -g_{ac}{J^c}_bwhere $$g_{ac} is the metric. That "J" and Ω usually have the same coordinate expression (up to an overall sign) is simply a consequence of the fact that the metric "g" is usually the identity matrix.

ee also

* symplectic vector space
* symplectic group
* symplectic representation
* orthogonal matrix
* unitary matrix
* Hamiltonian mechanics

External links

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