# Symplectic matrix

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Symplectic matrix

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

Properties

Every symplectic matrix is invertible with the inverse matrix given by:$M^\left\{-1\right\} = Omega^\left\{-1\right\} M^T Omega.$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 &plusmn;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:$mbox\left\{Pf\right\}\left(M^T Omega M\right) = det\left(M\right)mbox\left\{Pf\right\}\left(Omega\right).$Since $M^T Omega M = Omega$ and $mbox\left\{Pf\right\}\left(Omega\right) eq 0$ we have that det("M") = 1.

Suppose &Omega; is given in the standard form and let "M" be a 2"n"&times;2"n" block matrix given by:where "A, B, C, D" are "n"&times;"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&times;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 &omega; called the symplectic form.

A symplectic transformation is then a linear transformation "L" : "V" &rarr; "V" which preserves &omega;, i.e.:$omega\left(Lu, Lv\right) = omega\left(u, v\right).$Fixing a basis for "V", &omega; can be written as a matrix &Omega; 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:$Omega mapsto A^T Omega A$:$M mapsto A^\left\{-1\right\} M A.$One can always bring &Omega; 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 &Omega;. As explained in the previous section, &Omega; 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 &Omega; given above is the block diagonal form:This choice differs from the previous one by a permutation of basis vectors.

Sometimes the notation "J" is used instead of &Omega; 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 &Omega; but represents a very different structure. A complex structure "J" is the coordinate representation of a linear transformation that squares to −1, whereas &Omega; is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which "J" is not skew-symmetric or &Omega; does not square to −1.

Given a hermitian structure on a vector space, "J" and &Omega; are related via:$Omega_\left\{ab\right\} = -g_\left\{ac\right\}\left\{J^c\right\}_b$where $g_\left\{ac\right\}$ is the metric. That "J" and &Omega; 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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