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O que (quem) é symplectic collineation - definição

Symplectic transformation; Symplectic operator

Symplectic geometry

BRANCH OF DIFFERENTIAL GEOMETRY AND DIFFERENTIAL TOPOLOGY

Symplectic Geometry; Symplectic structure; Symplectic topology

Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry was founded by the Russian mathematician Vladimir Arnold and has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold.

https://en.wikipedia.org/wiki/Symplectic_geometry

Collineation

A BIJECTION FROM ONE PROJECTIVE SPACE TO ANOTHER THAT PRESERVES COLLINEARITY

Collineation group; Automorphic collineation; Colineation group

·noun The act of aiming at, or directing in a line with, a fixed object.

Collineation

A BIJECTION FROM ONE PROJECTIVE SPACE TO ANOTHER THAT PRESERVES COLLINEARITY

Collineation group; Automorphic collineation; Colineation group

In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus an isomorphism between projective spaces, or an automorphism from a projective space to itself.

https://en.wikipedia.org/wiki/Collineation

Wikipédia

Symplectic matrix

In mathematics, a symplectic matrix is a $2n\times 2n$ matrix $M$ with real entries that satisfies the condition

where $M^{\text{T}}$ denotes the transpose of $M$ and $\Omega$ is a fixed $2n\times 2n$ nonsingular, skew-symmetric matrix. This definition can be extended to $2n\times 2n$ matrices with entries in other fields, such as the complex numbers, finite fields, p-adic numbers, and function fields.

Typically $\Omega$ is chosen to be the block matrix

where

I_{n}

is the

n\times n

identity matrix. The matrix

\Omega

has determinant

+1

and its inverse is

\Omega ^{-1}=\Omega ^{\text{T}}=-\Omega

.

https://en.wikipedia.org/wiki/Symplectic matrix