Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a symplectic matrix is a 2nx2n matrix M (whose entries are typically either real or complex) satisfying the conditionwhere MT denotes the transpose of M and is a fixed nonsingular, skew-symmetric matrix. Typically is chosen to be the block matrixwhere In is the nxn identity matrix. Note that has determinant +1 and has an inverse given by –1 = T = – .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).

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