- 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 thetranspose of "M" and Ω is a fixed nonsingular,skew-symmetric matrix . Typically Ω is chosen to be theblock matrix :$Omega\; =egin\{bmatrix\}0\; I\_n\; \backslash -I\_n\; 0\; \backslash end\{bmatrix\}$where "I"_{n}is the "n"×"n"identity matrix . Note that Ω hasdeterminant +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 thesymplectic 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 thePfaffian 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:$M\; =\; egin\{pmatrix\}A\; B\; \backslash \; C\; Dend\{pmatrix\}$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 withlinear transformation s offinite-dimensional vector spaces . The abstract analog of a symplectic matrix is a**symplectic transformation**of asymplectic vector space . Briefly, a symplectic vector space is a 2"n"-dimensional vector space "V" equipped with anondegenerate ,skew-symmetric bilinear form ω called thesymplectic 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 anondegenerate skew-symmetric bilinear form . It is a basic result inlinear algebra that any two such matrices differ from each other by achange of basis .The most common alternative to the standard Ω given above is the

block diagonal form:$Omega\; =\; egin\{bmatrix\}egin\{matrix\}0\; 1\backslash \; -1\; 0end\{matrix\}\; 0\; \backslash \; ddots\; \backslash 0\; egin\{matrix\}0\; 1\; \backslash \; -1\; 0end\{matrix\}end\{bmatrix\}.$This choice differs from the previous one by apermutation 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:$Omega\_\{ab\}\; =\; -g\_\{ac\}\{J^c\}\_b$where $g\_\{ac\}$ 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.*