| dbo:abstract |
En matematiko, kvadrata formo super kampo F estas dirita al esti izotropa se estas ne-nula vektoro sur kiu ĝia valoro estas nulo. Alie la kvadrata formo estas neizotropa. Se q estas kvadrata formo sur vektora spaco V super F, tiam ne-nula vektoro v en V estas dirita al esti izotropa se q(v)=0. Kvadrata formo estas izotropa se kaj nur en V ekzistas ne-nula izotropa vektoro por la kvadrata formo. Estu (V, q) kvadrata spaco kaj W estu ĝia . Tiam W estas nomata kiel izotropa subspaco de V se ĉiuj vektoroj en ĝi estas izotropaj, kaj neizotropa subspaco se ĝi enhavas neniun (ne-nulan) izotropaj vektoroj. La izotropeca indekso de kvadrata spaco estas la maksimumo de la dimensioj de la izotropaj subspacoj. (eo) In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be isotropic if q(v) = 0. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form. Suppose that (V, q) is quadratic space and W is a subspace of V. Then W is called an isotropic subspace of V if some vector in it is isotropic, a totally isotropic subspace if all vectors in it are isotropic, and an anisotropic subspace if it does not contain any (non-zero) isotropic vectors. The isotropy index of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces. A quadratic form q on a finite-dimensional real vector space V is anisotropic if and only if q is a definite form: * either q is positive definite, i.e. q(v) > 0 for all non-zero v in V ; * or q is negative definite, i.e. q(v) < 0 for all non-zero v in V. More generally, if the quadratic form is non-degenerate and has the signature (a, b), then its isotropy index is the minimum of a and b. An important example of an isotropic form over the reals occurs in pseudo-Euclidean space. (en) 数学における等方二次形式（とうほうにじけいしき、英: isotropic quadratic form）は、ヌルベクトル（それに代入して零になるような非零ベクトル）を持つような二次形式を言う。等方的でない二次形式は非等方的 (anisotropic) と言う。 (ja) 在数学中，一个域 F 上的二次型称为迷向（isotropic）的如果在一个非零向量上取值为零。不然称为非迷向（anisotropic）的。更具体地，如果 q 是域 F 上向量空间 V 上一个二次型，则 V 中一个非零向量 v 称为迷向的如果 q(v)=0。一个二次型是迷向的当且仅当对这个二次型存在非零迷向向量。 假设 (V,q) 是二次空间，W 是一个子空间。如果 W 中所有向量都是迷向的，称之为 V 的一个迷向子空间；如果不存在任何非零迷向向量则称之为非迷向子空间。一个二次空间的迷向指标（isotropy index）是迷向子空间的最大维数。 (zh) |
| dbo:wikiPageExternalLink |
http://www.math.miami.edu/~armstrong/685fa12/pete_clark.pdf https://archive.org/details/courseinarithmet00serr |
| dbo:wikiPageID |
10917170 (xsd:integer) |
| dbo:wikiPageLength |
7178 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID |
1124978481 (xsd:integer) |
| dbo:wikiPageWikiLink |
dbr:Pseudo-Euclidean_space dbr:Definite_bilinear_form dbr:University_of_Miami dbr:Vector_space dbr:Null_vector dbr:Witt_group dbr:Quadratic_space dbc:Quadratic_forms dbr:Complex_numbers dbr:Coral_Gables,_Florida dbr:Chevalley–Warning_theorem dbr:Orthogonal dbr:Emil_Artin dbr:Signature_(quadratic_form) dbr:Plane_(geometry) dbr:Linear_subspace dbr:American_Mathematical_Society dbc:Bilinear_forms dbr:Field_(mathematics) dbr:Finite_field dbr:P-adic_number dbr:Graduate_Texts_in_Mathematics dbr:Isotropic_line dbr:Quadratic_form dbr:Hyperbola dbr:Hyperbolic_geometry dbr:Witt_ring_(forms) dbr:Symmetric_bilinear_form dbr:Polarization_identity dbr:Polynomial dbr:Hyperbolic-orthogonal dbr:Inner_product_space dbr:Algebraically_closed dbr:Orthogonal_complement dbr:Real_number dbr:Witt's_decomposition_theorem dbr:Linear_transformation dbr:Tsit_Yuen_Lam dbr:Polar_space dbr:Universal_quadratic_form dbr:Unit_hyperbola dbr:Springer-Verlag dbr:Orthogonal_direct_sum dbr:Characteristic_(field) dbr:W._A._Benjamin |
| dbp:wikiPageUsesTemplate |
dbt:Langle dbt:Rangle dbt:Cite_book dbt:Hatnote dbt:ISBN dbt:Reflist dbt:Rp dbt:Sfrac dbt:Short_description dbt:Visible_anchor |
| dcterms:subject |
dbc:Quadratic_forms dbc:Bilinear_forms |
| gold:hypernym |
dbr:Vector |
| rdf:type |
yago:WikicatBilinearForms yago:Abstraction100002137 yago:Form106290637 yago:LanguageUnit106284225 yago:Part113809207 yago:Relation100031921 yago:Word106286395 dbo:Insect yago:WikicatQuadraticForms |
| rdfs:comment |
数学における等方二次形式（とうほうにじけいしき、英: isotropic quadratic form）は、ヌルベクトル（それに代入して零になるような非零ベクトル）を持つような二次形式を言う。等方的でない二次形式は非等方的 (anisotropic) と言う。 (ja) 在数学中，一个域 F 上的二次型称为迷向（isotropic）的如果在一个非零向量上取值为零。不然称为非迷向（anisotropic）的。更具体地，如果 q 是域 F 上向量空间 V 上一个二次型，则 V 中一个非零向量 v 称为迷向的如果 q(v)=0。一个二次型是迷向的当且仅当对这个二次型存在非零迷向向量。 假设 (V,q) 是二次空间，W 是一个子空间。如果 W 中所有向量都是迷向的，称之为 V 的一个迷向子空间；如果不存在任何非零迷向向量则称之为非迷向子空间。一个二次空间的迷向指标（isotropy index）是迷向子空间的最大维数。 (zh) En matematiko, kvadrata formo super kampo F estas dirita al esti izotropa se estas ne-nula vektoro sur kiu ĝia valoro estas nulo. Alie la kvadrata formo estas neizotropa. Se q estas kvadrata formo sur vektora spaco V super F, tiam ne-nula vektoro v en V estas dirita al esti izotropa se q(v)=0. Kvadrata formo estas izotropa se kaj nur en V ekzistas ne-nula izotropa vektoro por la kvadrata formo. (eo) In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be isotropic if q(v) = 0. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form. A quadratic form q on a finite-dimensional real vector space V is anisotropic if and only if q is a definite form: (en) |
| rdfs:label |
Izotropa kvadrata formo (eo) Isotropic quadratic form (en) 等方二次形式 (ja) 迷向二次型 (zh) |
| owl:sameAs |
freebase:Isotropic quadratic form yago-res:Isotropic quadratic form wikidata:Isotropic quadratic form dbpedia-eo:Isotropic quadratic form dbpedia-ja:Isotropic quadratic form dbpedia-zh:Isotropic quadratic form https://global.dbpedia.org/id/4c89b |
| prov:wasDerivedFrom |
wikipedia-en:Isotropic_quadratic_form?oldid=1124978481&ns=0 |
| foaf:isPrimaryTopicOf |
wikipedia-en:Isotropic_quadratic_form |
| is dbo:wikiPageRedirects of |
dbr:Totally_isotropic dbr:Hyperbolic_plane_(quadratic_forms) dbr:Isotropy_index dbr:Anisotropic_quadratic_form dbr:Anisotropic_quadratic_space dbr:Anisotropic_subspace dbr:Totally_isotropic_subspace dbr:Isotropic_subspace dbr:Split_quadratic_space dbr:Metabolic_quadratic_space dbr:Isotropic_(quadratic_form) |
| is dbo:wikiPageWikiLink of |
dbr:Pseudo-Euclidean_space dbr:Totally_isotropic dbr:Degenerate_bilinear_form dbr:Hyperbolic_plane_(quadratic_forms) dbr:Vector_space dbr:Definite_quadratic_form dbr:List_of_irreducible_Tits_indices dbr:Null_vector dbr:Isotropy_index dbr:Norm_(mathematics) dbr:Glossary_of_linear_algebra dbr:Conformal_group dbr:Anisotropic_quadratic_form dbr:Anisotropic_quadratic_space dbr:Anisotropic_subspace dbr:Composition_algebra dbr:Polarization_(Lie_algebra) dbr:Cayley–Dickson_construction dbr:Totally_isotropic_subspace dbr:Hasse–Minkowski_theorem dbr:Linked_field dbr:Isometry_group dbr:Isotropic_line dbr:Quadratic_form dbr:Quasi-algebraically_closed_field dbr:Isotropic_subspace dbr:Isotropy dbr:Biquaternion_algebra dbr:Split-octonion dbr:Special_unitary_group dbr:Spin_representation dbr:Split-complex_number dbr:Split-quaternion dbr:Split_quadratic_space dbr:Metabolic_quadratic_space dbr:Seminorm dbr:Hyperbolic dbr:Octonion_algebra dbr:Polar_space dbr:U-invariant dbr:Pfister_form dbr:Spread_(projective_geometry) dbr:Isotropic_(quadratic_form) |
| is rdfs:seeAlso of |
dbr:Quadratic_form |
| is foaf:primaryTopic of |
wikipedia-en:Isotropic_quadratic_form |