Symplectic matrix
In mathematics, a symplectic matrix is a 2n by 2n matrix M (whose entries are typically either real or complex) satisfying the condition
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where MT denotes the transpose of M and J is the 2n×2n skew-symmetric matrix
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(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
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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. Though harder to prove, it can actually be shown that the determinant of a symplectic matrix is always +1.
Let M be a 2n×2n block matrix given by
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where A, B, C, D are n×n matrices. Then the condition for M to be symplectic is equivalent to the conditions
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See also: orthogonal matrix, unitary matrix, symplectic space
