Symplectic matrix

In mathematics, a symplectic matrix is a 2n × 2n matrix M with real entries that satisfies the condition M T Ω M = Ω , {displaystyle M^{ ext{T}}Omega M=Omega ,}     (1) S = O ( D 0 0 D − 1 ) O ′ , {displaystyle S=O{egin{pmatrix}D&0\0&D^{-1}end{pmatrix}}O',}     (2) M ∗ Ω M = Ω . {displaystyle M^{*}Omega M=Omega ,.}     (3) In mathematics, a symplectic matrix is a 2n × 2n matrix M with real entries that satisfies the condition where MT denotes the transpose of M and Ω is a fixed 2n × 2n nonsingular, skew-symmetric matrix. This definition can be extended to 2n × 2n matrices with entries in other fields, such as the complex numbers. Typically Ω is chosen to be the block matrix where In 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 determinant 1, and the 2n × 2n symplectic matrices with real entries form a subgroup of the special linear group SL(2n, R) under matrix multiplication. Topologically, this symplectic group is a connected noncompact real Lie group of real dimension n(2n + 1), and is denoted Sp(2n, R). The symplectic group can be defined as the set of linear transformations that preserve the symplectic form of a real symplectic vector space. Examples of symplectic matrices include the identity matrix and the matrix [ 1 1 0 1 ] {displaystyle {egin{bmatrix}1&1\0&1end{bmatrix}}} . Every symplectic matrix is invertible with the inverse matrix given by 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 for any field. One way to see this is through the use of the Pfaffian and the identity