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Symplectic MatrixIn mathematics, a symplectic matrix is a 2n by 2n matrix M (whose entries are typically either real or complex) satisfying the condition -
where MT denotes the transpose of M and J is the 2n×2n skew-symmetric matrix -
\begin{pmatrix} 0 & I_n \\ -I_n & 0 \\ \end{pmatrix} (In being the n×n identity matrix). Note that J has determinant +1 and squares to minus the identity: J2 = −I2n. Every symplectic matrix has an inverse which is 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. The symplectic group has dimension n(2n + 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 -
Since we have that det(M) = 1. Let M be a 2n×2n block matrix given by -
where A, B, C, D are n×n matrices. Then the condition for M to be symplectic is equivalent to the conditions -
-
-
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. Related topics
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