Symplectic matrix


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 "MT" 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

*
*


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Symplectic vector space — In mathematics, a symplectic vector space is a vector space V equipped with a nondegenerate, skew symmetric, bilinear form omega; called the symplectic form. Explicitly, a symplectic form is a bilinear form omega; : V times; V rarr; R which is *… …   Wikipedia

  • Symplectic manifold — In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2 form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology.… …   Wikipedia

  • Symplectic group — For finite groups with all characteristc abelian subgroups cyclic, see group of symplectic type. Group theory …   Wikipedia

  • Hamiltonian matrix — In mathematics, a Hamiltonian matrix A is any real 2n×2n matrix that satisfies the condition that KA is symmetric, where K is the skew symmetric matrix:K=egin{bmatrix}0 I n I n 0 end{bmatrix}and In is the n×n identity matrix. In other words, A… …   Wikipedia

  • Orthogonal matrix — In linear algebra, an orthogonal matrix (less commonly called orthonormal matrix[1]), is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors). Equivalently, a matrix Q is orthogonal if… …   Wikipedia

  • Skew-symmetric matrix — In linear algebra, a skew symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation:: A T = − A or in component form, if A = ( a ij ):: a ij = − a ji for all i and j .For… …   Wikipedia

  • Unitary matrix — In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition:U^* U = UU^* = I n, where I n, is the identity matrix and U^* , is the conjugate transpose (also called the Hermitian adjoint) of U . Note this condition says …   Wikipedia

  • Random matrix — In probability theory and mathematical physics, a random matrix is a matrix valued random variable. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a… …   Wikipedia

  • Skew-Hamiltonian matrix — In linear algebra, skew Hamiltonian matrices are special matrices which correspond to skew symmetric bilinear forms on a symplectic vector space.Let V be a vector space, equipped with a symplectic form Omega. Such a space must be even dimensional …   Wikipedia

  • Unimodular matrix — In mathematics, a unimodular matrix M is a square integer matrix with determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N which is its inverse (these are equivalent under… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.