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Concept
Symplectic matrix
Summary
In mathematics, a symplectic matrix is a matrix with real entries that satisfies the condition
where denotes the transpose of and is a fixed nonsingular, skew-symmetric matrix. This definition can be extended to matrices with entries in other fields, such as the complex numbers, finite fields, p-adic numbers, and function fields.
Typically is chosen to be the block matrix
where is the identity matrix. The matrix has determinant and its inverse is .
Every symplectic matrix has determinant , and the symplectic matrices with real entries form a subgroup of the general linear group under matrix multiplication since being symplectic is a property stable under matrix multiplication. Topologically, this symplectic group is a connected noncompact real Lie group of real dimension , and is denoted . The symplectic group can be defined as the set of linear transformations that preserve the symplectic form of a real symplectic vector space.
This symplectic group has a distinguished set of generators, which can be used to find all possible symplectic matrices. This includes the following sets
where is the set of symmetric matrices. Then, is generated by the setp. 2
of matrices. In other words, any symplectic matrix can be constructed by multiplying matrices in and together, along with some power of .
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
Since and we have that .
When the underlying field is real or complex, one can also show this by factoring the inequality .
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