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- Symplectic_vector_space abstract "In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers R) equipped with a symplectic bilinear form ω : V × V → F. The bilinear form ω is said to be symplectic, if it is Alternating: ω(v, v) = 0 for all v ∈ V, and Nondegenerate: if ω(u, v) = 0 for all v ∈ V then u = 0.If the underlying field has characteristic ≠ 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa. Working in a fixed basis, ω can be represented by a matrix. The conditions above say that this matrix must be skew-symmetric, nonsingular, and hollow. This is not the same thing as a symplectic matrix, which represents a symplectic transformation of the space. If V is finite-dimensional, then its dimension must necessarily be even since every skew-symmetric, hollow matrix of odd size has determinant zero. Notice the condition that the matrix be hollow is not redundant if the characteristic of the field is 2. A symplectic form behaves quite differently from a symmetric form, such as the dot product on Euclidean vector spaces. With a Euclidean inner product g, we have g(v,v) > 0 for all nonzero vectors v.".
- Symplectic_vector_space wikiPageID "292852".
- Symplectic_vector_space wikiPageRevisionID "604116956".
- Symplectic_vector_space hasPhotoCollection Symplectic_vector_space.
- Symplectic_vector_space subject Category:Bilinear_forms.
- Symplectic_vector_space subject Category:Linear_algebra.
- Symplectic_vector_space subject Category:Symplectic_geometry.
- Symplectic_vector_space type Abstraction100002137.
- Symplectic_vector_space type BilinearForms.
- Symplectic_vector_space type Form106290637.
- Symplectic_vector_space type LanguageUnit106284225.
- Symplectic_vector_space type Part113809207.
- Symplectic_vector_space type Relation100031921.
- Symplectic_vector_space type Word106286395.
- Symplectic_vector_space comment "In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers R) equipped with a symplectic bilinear form ω : V × V → F. The bilinear form ω is said to be symplectic, if it is Alternating: ω(v, v) = 0 for all v ∈ V, and Nondegenerate: if ω(u, v) = 0 for all v ∈ V then u = 0.If the underlying field has characteristic ≠ 2, alternation is equivalent to skew-symmetry.".
- Symplectic_vector_space label "Espace vectoriel symplectique".
- Symplectic_vector_space label "Espacio vectorial simpléctico".
- Symplectic_vector_space label "Spazio vettoriale simplettico".
- Symplectic_vector_space label "Symplectic vector space".
- Symplectic_vector_space label "Symplektischer Vektorraum".
- Symplectic_vector_space label "Симплектическое пространство".
- Symplectic_vector_space label "斜交ベクトル空間".
- Symplectic_vector_space label "辛向量空间".
- Symplectic_vector_space sameAs Symplektický_vektorový_prostor.
- Symplectic_vector_space sameAs Symplektischer_Vektorraum.
- Symplectic_vector_space sameAs Espacio_vectorial_simpléctico.
- Symplectic_vector_space sameAs Espace_vectoriel_symplectique.
- Symplectic_vector_space sameAs Spazio_vettoriale_simplettico.
- Symplectic_vector_space sameAs 斜交ベクトル空間.
- Symplectic_vector_space sameAs m.01qtjt.
- Symplectic_vector_space sameAs Q766774.
- Symplectic_vector_space sameAs Q766774.
- Symplectic_vector_space sameAs Symplectic_vector_space.
- Symplectic_vector_space wasDerivedFrom Symplectic_vector_space?oldid=604116956.
- Symplectic_vector_space isPrimaryTopicOf Symplectic_vector_space.