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In mathematics, a symplectic matrix is a 2n×2n matrix M (whose entries are typically either real or complex) satisfying the condition Mathematics is the study of quantity, structure, space and change. ...
For the square matrix section, see square matrix. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...
- MTΩM = Ω.
where MT denotes the transpose of M and Ω is the 2n×2n skew-symmetric matrix See transposition for meanings of this term in telecommunication and music. ...
In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji for all i and j. ...
Here In is the n×n identity matrix. Note that Ω has determinant +1 and squares to minus the identity: Ω2 = −I2n. In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...
In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
N.B. Some authors prefer to use a different Ω for the definition of symplectic matrices. The only essential property is that Ω be a nonsingular, skew-symmetric matrix. The most common alternative is the block diagonal form In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...
In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji for all i and j. ...
In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks. ...
Note that this differs from the previous choice by a permutation of basis vectors. In fact, any choice of Ω can be brought to either of the above forms by a different choice of basis. See the abstract formulation below in the section on symplectic transformations. In mathematics, especially in abstract algebra and related areas, a permutation is a bijection, from a finite set X onto itself. ...
Properties
Every symplectic matrix has an inverse which is given by In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...
- M − 1 = Ω − 1MTΩ
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). In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ...
In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. ...
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 In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries. ...
- Pf(MTΩM) = det(M)Pf(Ω).
Since MTΩM = Ω and we have that det(M) = 1.
Let M be a 2n×2n block matrix given by In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks. ...
where A, B, C, D are n×n matrices. Then the condition for M to be symplectic is equivalent to the conditions - ATD − CTB = 1
- ATC = CTA
- DTB = BTD.
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. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include Q is necessary and sufficient for P and P...
Symplectic transformations In the abstract formulation of linear algebra, matrices are replaced with linear transformations of finite-dimensional vector spaces. The abstract analog of a symplectic matrix is a symplectic transformation of a symplectic vector space. Briefly, a symplectic vector space is a 2n-dimensional vector space V equipped with a nondegenerate, skew-symmetric bilinear form ω. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
In mathematics, the dimension of a vector space V is the cardinality (i. ...
The fundamental concept in linear algebra is that of a vector space or linear space. ...
In mathematics, a symplectic vector space is a vector space V equipped with a nondegenerate, skew-symmetric, bilinear form ω called the symplectic form. ...
In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ...
In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji for all i and j. ...
In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ...
A symplectic transformation is then a linear transformation f : V → V which preserves ω, i.e.
- ω(f(x),f(y)) = ω(x,y).
Fixing a basis for V, ω can be written as a matrix Ω and f as a matrix M. The condition that f be a symplectic transformation is precisely the condition that M be a symplectic matrix: In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...
- MTΩM = Ω.
Under a change of basis, represented by a matrix A, we have In linear algebra, we may consider some finite dimensional vector space, which can have associated with it some basis with which we can work with respect to. ...
One can always bring Ω to either of the standard forms given in the introduction by a suitable choice of A.
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