In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function that maps every pair of elements of the vector space to the underlying field such that for every and in . They are also referred to more briefly as just symmetric forms when "bilinear" is understood.
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| - Simetria dulineara formo (eo)
- Forme bilinéaire symétrique (fr)
- 対称双線型形式 (ja)
- Symmetrische bilineaire vorm (nl)
- Forma bilinear simétrica (pt)
- Symmetric bilinear form (en)
- 对称双线性形式 (zh)
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| - En matematiko, simetria dulineara formo sur vektora spaco estas dualineara bildigo de du kopioj de la vektora spaco al la kampo de skalaroj tia ke la ordo de la du vektoroj ne influas la valoron de la bildigo. Alivorte, ĝi estas dualineara funkcio kiu mapas ĉiun paron da elementoj de la vektora spaco al la suba kampo tia ke por ĉiuj kaj en . Ili ankaŭ estas nomataj pli koncize kiel nur simetriaj formoj kiam "dulineara" estas komprenita. (eo)
- En algèbre linéaire, une forme bilinéaire symétrique est une forme bilinéaire qui a la particularité d'être symétrique. Les formes bilinéaires symétriques jouent un rôle important dans l'étude des quadriques. (fr)
- 線型代数学における対称双線型形式（たいしょうそうせんけいけいしき、英: symmetric bilinear form, symmetric bilinear functional）は、ベクトル空間上の対称な双線型形式を言う。平たく言えば、実ベクトル空間上の標準内積を一般化した概念である。対称双線型形式は、直交極性や二次曲面の研究に非常に重要である。 文脈上、双線型形式について述べていると明らかな場合は、単に短く対称形式と呼ぶこともある。対称双線型形式は二次形式と近しい関係にあり、この両者の差異に関する詳細はの項目を参照。 (ja)
- Uma forma bilinear simétrica em um espaço vetorial V sobre um corpo K é uma função satisfazendo:
* B é uma forma bilinear, ou seja
*
*
*
* B é simétrica, ou seja
* Formas bilineares simétricas são importantes no estudo das quádricas e na teoria da relatividade, em que o "produto interno" é uma forma bilinear simétrica não-degenerada. (pt)
- 对称双线性形式是在向量空间上的对称双线性形式。它们在正交极性和的研究中非常重要。 (zh)
- In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function that maps every pair of elements of the vector space to the underlying field such that for every and in . They are also referred to more briefly as just symmetric forms when "bilinear" is understood. (en)
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| - Symmetric Bilinear Form (en)
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| - SymmetricBilinearForm (en)
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| - En matematiko, simetria dulineara formo sur vektora spaco estas dualineara bildigo de du kopioj de la vektora spaco al la kampo de skalaroj tia ke la ordo de la du vektoroj ne influas la valoron de la bildigo. Alivorte, ĝi estas dualineara funkcio kiu mapas ĉiun paron da elementoj de la vektora spaco al la suba kampo tia ke por ĉiuj kaj en . Ili ankaŭ estas nomataj pli koncize kiel nur simetriaj formoj kiam "dulineara" estas komprenita. (eo)
- En algèbre linéaire, une forme bilinéaire symétrique est une forme bilinéaire qui a la particularité d'être symétrique. Les formes bilinéaires symétriques jouent un rôle important dans l'étude des quadriques. (fr)
- In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function that maps every pair of elements of the vector space to the underlying field such that for every and in . They are also referred to more briefly as just symmetric forms when "bilinear" is understood. Symmetric bilinear forms on finite-dimensional vector spaces precisely correspond to symmetric matrices given a basis for V. Among bilinear forms, the symmetric ones are important because they are the ones for which the vector space admits a particularly simple kind of basis known as an orthogonal basis (at least when the characteristic of the field is not 2). Given a symmetric bilinear form B, the function q(x) = B(x, x) is the associated quadratic form on the vector space. Moreover, if the characteristic of the field is not 2, B is the unique symmetric bilinear form associated with q. (en)
- 線型代数学における対称双線型形式（たいしょうそうせんけいけいしき、英: symmetric bilinear form, symmetric bilinear functional）は、ベクトル空間上の対称な双線型形式を言う。平たく言えば、実ベクトル空間上の標準内積を一般化した概念である。対称双線型形式は、直交極性や二次曲面の研究に非常に重要である。 文脈上、双線型形式について述べていると明らかな場合は、単に短く対称形式と呼ぶこともある。対称双線型形式は二次形式と近しい関係にあり、この両者の差異に関する詳細はの項目を参照。 (ja)
- Uma forma bilinear simétrica em um espaço vetorial V sobre um corpo K é uma função satisfazendo:
* B é uma forma bilinear, ou seja
*
*
*
* B é simétrica, ou seja
* Formas bilineares simétricas são importantes no estudo das quádricas e na teoria da relatividade, em que o "produto interno" é uma forma bilinear simétrica não-degenerada. (pt)
- 对称双线性形式是在向量空间上的对称双线性形式。它们在正交极性和的研究中非常重要。 (zh)
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