Orthogonal group (original) (raw)
Ortogonální grupa je množina všech rotací a zrcadlení Euklidova prostoru spolu s operací skládání. Obecněji jde o grupu lineárních transformací vektorového prostoru zachovávajících nějakou symetrickou bilineární formu.
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| dbo:abstract | En matemàtiques, el grup ortogonal de dimensió n, denotat O(n), és el grup de transformacions isomètriques (que preserven la distància) d'un espai Euclidià de dimensió n que preserven un punt fix, on l'operació de grup és donada per la composició de transformacions. De forma equivalent, és el grup de matrius ortogonals n×n (és a dir matrius per les quals la seva matriu inversa és igual a la seva transposada), i on l'operació de grup és donada per la multiplicació matricial. Com que el determinant d'una matriu O(n) ortogonal és 1 o −1, un subgrup important de O(n) és el grup ortogonal especial, denotat SO(n), de les matrius ortogonals amb determinant 1. Aquest grup és també anomenat grup de rotació, perquè en dimensions 2 i 3, els seus elements són les rotacions habituals al voltant d'un punt (dimensió 2) o una línia (dimensió 3). A petites dimensions, com SO(2), SO(3) i SO(4), aquests grups han estat àmpliament estudiats. En física de partícules, els grups ortonormals tenen un paper important en la categorització de les forces i partícules elementals. El primer grup de Lie simple que conté el Model estàndard és SO(10), sent el grup més senzill que aconsegueix la unificació de totes les partícules de matèria, incloent-hi els neutrins. (ca) في الرياضيات، زمرة متعامدة (بالإنجليزية: Orthogonal group) بُعدها هو n، هي زمرة التحويلات المحافظة على المسافة في الفضاء الإقليدي اللائي يحافظن على نقطة ثابتة معينة. العملية المعرِّفة لهذه الزمرة هي تركيب التحويلات. يرمز إليها ب O(n). الزمرة المتعامدة بالنسبة للأعداد الحقيقة يمكن كتابتها بشكل عام ك: انظر إلى مصفوفة متعامدة وإلى مصفوفة قابلة للعكس وإلى منقولة مصفوفة. هناك أيضا زمرة جزيئية من الزمرة المتعامدة مهمة أيضا في الفيزياء، ويطلق عليها الزمرة المتعامدة الخاصة (Special Orthogonal Group) و يرمز لها ب و تمثل بالنسبة للأعداد الحقيقية مجموعة المصفوفات المتعادمة التدويرية، التي تقوم بتدوير أي متجه في الفضاء الإقليدي بزاوية معينة دون تغيير طوله: (ar) Ortogonální grupa je množina všech rotací a zrcadlení Euklidova prostoru spolu s operací skládání. Obecněji jde o grupu lineárních transformací vektorového prostoru zachovávajících nějakou symetrickou bilineární formu. (cs) Die orthogonale Gruppe ist die Gruppe der orthogonalen -Matrizen mit reellen Elementen. Die Verknüpfung der orthogonalen Gruppe ist die Matrizenmultiplikation. Bei der orthogonalen Gruppe handelt es sich um eine Lie-Gruppe der Dimension .Da die Determinante einer orthogonalen Matrix nur die Werte annehmen kann, zerfällt in die beiden disjunkten Teilmengen (topologisch: Zusammenhangskomponenten) * die Drehgruppe aller Drehungen (orthogonale Matrizen mit Determinante ) und * aller Drehspiegelungen (orthogonale Matrizen mit Determinante ). Die Untergruppe heißt die spezielle orthogonale Gruppe. Insbesondere ist die als die Gruppe aller Drehungen um eine durch den Koordinatenursprung verlaufende Achse im dreidimensionalen Raum von großer Bedeutung in zahlreichen Anwendungen, wie etwa der Computergraphik oder der Physik. (de) En matemática, el grupo ortogonal de grado n sobre un cuerpo , designado como , es el grupo de matrices ortogonales n por n con las entradas en , con la operación de grupo dada por la multiplicación de matrices. Este es un subgrupo del grupo general lineal . Cada matriz ortogonal tiene determinante 1 o -1. Las matrices n por n ortogonales con determinante 1 forman un subgrupo normal de conocido como el grupo ortogonal especial , también conocido como grupo de rotaciones. Si la característica de es 2, entonces y coinciden; en caso contrario el índice de en es 2. y son , porque la condición que una matriz sea ortogonal, es decir que su propia transpuesta sea su inversa, se puede expresar como un conjunto de ecuaciones polinómicas en las entradas de la matriz. (es) In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO(n). It consists of all orthogonal matrices of determinant 1. This group is also called the rotation group, generalizing the fact that in dimensions 2 and 3, its elements are the usual rotations around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see SO(2), SO(3) and SO(4). The other component consists of all orthogonal matrices of determinant –1. This component does not form a group, as the product of any two of its elements is of determinant 1, and therefore not an element of the component. By extension, for any field F, an n×n matrix with entries in F such that its inverse equals its transpose is called an orthogonal matrix over F. The n×n orthogonalmatrices form a subgroup, denoted O(n, F), of the general linear group GL(n, F); that is More generally, given a non-degenerate symmetric bilinear form or quadratic form on a vector space over a field, the orthogonal group of the form is the group of invertible linear maps that preserve the form. The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the dot product, or, equivalently, the quadratic form is the sum of the square of the coordinates. All orthogonal groups are algebraic groups, since the condition of preserving a form can be expressed as an equality of matrices. (en) En mathématiques, le groupe orthogonal est formé de transformations géométriques préservant les distances (isométries) et le point origine de l'espace. Formellement, on introduit le groupe orthogonal d'une forme quadratique q sur E, espace vectoriel sur un corps commutatif K, comme le sous-groupe du groupe linéaire GL(E) constitué des automorphismes f de E qui laissent q invariante : q(f(x)) = q(x) pour tout vecteur x de E.La loi de composition de ce groupe est la composition des applications. Dans cet article, K désigne un corps commutatif et E un espace vectoriel de dimension finie non nulle n sur K et q désigne une forme quadratique non dégénérée sur E. (fr) In matematica, il gruppo ortogonale di grado su un campo è il gruppo delle matrici ortogonali a valori in . Si indica con o, se il campo è chiaro dal contesto, semplicemente con . Quando è il campo dei numeri reali, il gruppo può essere interpretato come il gruppo delle isometrie dello spazio euclideo di dimensione Le matrici aventi determinante uguale a formano un sottogruppo, che si indica con , detto gruppo ortogonale speciale. Il gruppo ortogonale speciale è il gruppo delle rotazioni dello spazio. (it) 군론에서 직교군(直交群, 영어: orthogonal group)은 주어진 체에 대한 직교 행렬의 리 군이다. (ko) In de wiskunde is de orthogonale groep van over een lichaam (Ned) / veld (Be) , genoteerd als , de groep van isometrieën in de -dimensionale ruimte die de oorsprong vast houden. De orthogonale groep komt overeen met de groep van orthogonale -matrices met elementen uit men als groepsbewerking de matrixvermenigvuldiging. Dit is een ondergroep van de algemene lineaire groep bepaald door waarin de getransponeerde van is. De klassieke orthogonale groep over de reële getallen wordt meestal als geschreven. Meer in het algemeen is de orthogonale groep van een niet-singuliere kwadratische vorm over de groep van matrices die deze kwadratische vorm bewaart. De stelling van Cartan-Dieudonné beschrijft de wiskundige structuur van de orthogonale groep. Elke orthogonale matrix heeft een determinant die of gelijk is aan 1 of gelijk is aan −1. De orthogonale -matrices met determinant 1 vormen een normaaldeler van , die bekendstaat als de speciale orthogonale groep . Als de karakteristiek van gelijk is aan 2, geldt dat 1 = −1, en vallen en dus samen, anders is de nevenklasse van in gelijk aan 2. In karakteristiek 2 en met even dimensie definiëren vele auteurs alternatief als de kern van de ; dan heeft deze kern meestal de index 2 in . Zowel als zijn algebraïsche groepen, omdat de voorwaarde dat een matrix orthogonaal moet zijn, dat wil zeggen dat een matrix zijn eigen getransponeerde als inverse moet hebben, kan worden uitgedrukt als een verzameling van polynomiale vergelijkingen in de elementen van de matrix. (nl) 数学において、n 次元の直交群（ちょっこうぐん、英: orthogonal group）とは、n 次元ユークリッド空間上のある固定された点を保つような距離を保つ変換全体からなる群であり、群の演算は変換の合成によって与える。O(n) と表記する。同値な別の定義をすれば、直交群とは、元がn×n の実直交行列であり、群の積が行列の積によって与えられるものをいう。直交行列とは、逆行列がもとの行列の転置と等しくなるような行列のことである。 直交行列の行列式は 1 か −1 である。O(n) の重要な部分群である特殊直交群 SO(n) は行列式が 1 である直交行列からなる。この群は回転群ともよばれ、例えば次元 2 や 3 では、群の元が表す変換は（2次元における）点や（3次元における）直線のまわりの通常の回転である。低次元ではこれらの群の性質は幅広く研究されている。 用語「直交群」は上の定義を一般化して、体上のベクトル空間における非退化な対称双線型形式や二次形式を保つような、可逆な線形作用素全体からなる群を表すことがある。特に、体 F 上の n 次元ベクトル空間 F n 上の双線型形式がドット積で与えられ、二次形式が二乗の和で与えられるとき、これに対応する直交群 O(n, F) は、群の元が F 成分 n × n 直交行列で群の積を行列の積で定めるものである。これは一般線形群 GL(n, F ) の部分群であって、以下の形で与えられる。 ここで QT は Q の転置であり、 I は単位行列である。 (ja) En ortogonalgrupp är ett matematisk begrepp inom linjär algebra. Ortogonalgruppen är en grupp bestående av linjära avbildningar med egenskapen att de bevarar skalärprodukten. Ortogonalgruppen är en undergrupp till den (sv) Ортогональная группа — группа всех линейных преобразований -мерного векторного пространства надполем , сохраняющих фиксированную невырожденную квадратичную форму на (то есть таких линейных преобразований , что для любого ). (ru) Ортогональна група розміру n — група перетворень евклідового простору розмірності n, які зберігають відстані. Може бути описана групою ортогональних матриць розміру n×n відносно операції множення. Позначається O(n). Визначник ортогональної матриці може дорівнювати 1 чи -1. Важливим частковим випадком ортогональної групи є спеціальна ортогональна група — група ортогональних матриць із визначником 1 (група матриць повороту, позначається SO(n). Ця група також називається групою обертань, оскільки для розмірності 2 її елементи є обертанням навколо точки, а для розмірності 3 — обертанням навколо осі. Для малих розмірностей ці групи широко застосовуються: , SO(3), SO(4). (uk) Em matemática, um grupo ortogonal é um grupo de todas as transformações lineares de um espaço vetorial de dimensões de um campo, que preserva a um não singular fixo de forma quadrática em , (ou seja, as transformações lineares tal que para todos ).Um grupo ortogonal é um . Os elementos de um grupo ortogonal são chamados transformações ortogonais de (com relação a ), ou também de automorfismos de forma . Além disso, permita (para grupos ortogonais sobre os campos com característica 2 e deixe ser a forma bilinear simétrica não singular em relacionada com o pela fórmula O grupo ortogonal, então, consiste naqueles transformações lineares de V que preservam f, e é indicado por ou (quando está se falando de um campo específico e uma forma específica ) simplesmente por . Se é a matriz de em relação a algumas bases de , então o grupo ortogonal pode ser identificado com o grupo de todos os -matrizes A com coeficientes de tal que (onde representa a matriz transposta). O determinante de uma matriz ortogonal sendo 1 ou -1, um subgrupo importante de é o grupo especial ortogonal, denotado , das matrizes ortogonais do determinante 1. (pt) 数学上，数域F上的n阶正交群，记作O(n,F)，是F上的n×n 正交矩阵在矩阵乘法下构成的群。它是一般线性群GL(n,F)的子群，由 给出。 这里QT是Q的转置。实数域上的经典正交群通常就记为O(n)。 更一般地，F上一个非奇异二次型的正交群是保持二次型不变的矩阵构成的群。嘉当-迪奥多内定理描述了这个正交群的结构。 每个正交矩阵的行列式为1或−1。行列式为1的n×n正交矩阵组成一个O(n,F)的正规子群，称为特殊正交群SO(n,F)。如果F的特徵为2，那么1 = −1，从而O(n,F)和SO(n,F)相等；其他情形SO(n,F)在O(n,F)中的指数是2。特征2且偶数维时，很多作者用另一种定义，定义SO(n,F)为迪克森不变量的核，这样它在O(n,F)中总有指数2。 O(n,F)和SO(n,F)都是代数群，因为如果一个矩阵是正交的条件，即转置等于逆矩阵，能够定义成一些关于矩阵分量的多项式方程。 (zh) |
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| dbp:date | May 2020 (en) November 2019 (en) |
| dbp:drop | hidden (en) |
| dbp:id | p/o070300 (en) |
| dbp:proof | For studying the orthogonal group of , one can suppose that the matrix of the quadratic form is because, given a quadratic form, there is a basis where its matrix is diagonalizable. A matrix belongs to the orthogonal group if that is, , , and . As and cannot be both zero , the second equation implies the existence of in , such that and . Reporting these values in the third equation, and using the first equation, one gets that , and thus the orthogonal group consists of the matrices : where and . Moreover, the determinant of the matrix is . For further studying the orthogonal group, it is convenient to introduce a square root of . This square root belongs to if the orthogonal group is , and to otherwise. Setting , and , one has : If and are two matrices of determinant one in the orthogonal group then : This is an orthogonal matrix with , and . Thus : It follows that the map is a homomorphism of the group of orthogonal matrices of determinant one into the multiplicative group of . In the case of , the image is the multiplicative group of , which is a cyclic group of order . In the case of , the above and are conjugate, and are therefore the image of each other by the Frobenius automorphism. This meant that and thus For every such one can reconstruct a corresponding orthogonal matrix. It follows that the map is a group isomorphism from the orthogonal matrices of determinant 1 to the group of the -roots of unity. This group is a cyclic group of order which consists of the powers of where is a primitive element of , For finishing the proof, it suffices to verify that the group all orthogonal matrices is not abelian, and is the semidirect product of the group and the group of orthogonal matrices of determinant one. The comparison of this proof with the real case may be illuminating. Here two group isomorphisms are involved: : where is a primitive element of and is the multiplicative group of the element of norm one in ; : with and In the real case, the corresponding isomorphisms are: : where is the circle of the complex numbers of norm one; : with and (en) |
| dbp:reason | most notations are undefined; no context for explaining why these consideration belong to the article. Moreover, the section consists essentially in a list of advanced results without providing the information that is needed for a non-specialist for verifying them (en) This seems like it is the associated polar form B′=Q−Q−Q . When expressed in the same terms , the expression for a reflection is the same for all cases. (en) |
| dbp:title | Orthogonal group (en) Proof: (en) |
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| rdfs:comment | Ortogonální grupa je množina všech rotací a zrcadlení Euklidova prostoru spolu s operací skládání. Obecněji jde o grupu lineárních transformací vektorového prostoru zachovávajících nějakou symetrickou bilineární formu. (cs) In matematica, il gruppo ortogonale di grado su un campo è il gruppo delle matrici ortogonali a valori in . Si indica con o, se il campo è chiaro dal contesto, semplicemente con . Quando è il campo dei numeri reali, il gruppo può essere interpretato come il gruppo delle isometrie dello spazio euclideo di dimensione Le matrici aventi determinante uguale a formano un sottogruppo, che si indica con , detto gruppo ortogonale speciale. Il gruppo ortogonale speciale è il gruppo delle rotazioni dello spazio. (it) 군론에서 직교군(直交群, 영어: orthogonal group)은 주어진 체에 대한 직교 행렬의 리 군이다. (ko) En ortogonalgrupp är ett matematisk begrepp inom linjär algebra. Ortogonalgruppen är en grupp bestående av linjära avbildningar med egenskapen att de bevarar skalärprodukten. Ortogonalgruppen är en undergrupp till den (sv) Ортогональная группа — группа всех линейных преобразований -мерного векторного пространства надполем , сохраняющих фиксированную невырожденную квадратичную форму на (то есть таких линейных преобразований , что для любого ). (ru) 数学上，数域F上的n阶正交群，记作O(n,F)，是F上的n×n 正交矩阵在矩阵乘法下构成的群。它是一般线性群GL(n,F)的子群，由 给出。 这里QT是Q的转置。实数域上的经典正交群通常就记为O(n)。 更一般地，F上一个非奇异二次型的正交群是保持二次型不变的矩阵构成的群。嘉当-迪奥多内定理描述了这个正交群的结构。 每个正交矩阵的行列式为1或−1。行列式为1的n×n正交矩阵组成一个O(n,F)的正规子群，称为特殊正交群SO(n,F)。如果F的特徵为2，那么1 = −1，从而O(n,F)和SO(n,F)相等；其他情形SO(n,F)在O(n,F)中的指数是2。特征2且偶数维时，很多作者用另一种定义，定义SO(n,F)为迪克森不变量的核，这样它在O(n,F)中总有指数2。 O(n,F)和SO(n,F)都是代数群，因为如果一个矩阵是正交的条件，即转置等于逆矩阵，能够定义成一些关于矩阵分量的多项式方程。 (zh) في الرياضيات، زمرة متعامدة (بالإنجليزية: Orthogonal group) بُعدها هو n، هي زمرة التحويلات المحافظة على المسافة في الفضاء الإقليدي اللائي يحافظن على نقطة ثابتة معينة. العملية المعرِّفة لهذه الزمرة هي تركيب التحويلات. يرمز إليها ب O(n). الزمرة المتعامدة بالنسبة للأعداد الحقيقة يمكن كتابتها بشكل عام ك: انظر إلى مصفوفة متعامدة وإلى مصفوفة قابلة للعكس وإلى منقولة مصفوفة. (ar) En matemàtiques, el grup ortogonal de dimensió n, denotat O(n), és el grup de transformacions isomètriques (que preserven la distància) d'un espai Euclidià de dimensió n que preserven un punt fix, on l'operació de grup és donada per la composició de transformacions. De forma equivalent, és el grup de matrius ortogonals n×n (és a dir matrius per les quals la seva matriu inversa és igual a la seva transposada), i on l'operació de grup és donada per la multiplicació matricial. (ca) Die orthogonale Gruppe ist die Gruppe der orthogonalen -Matrizen mit reellen Elementen. Die Verknüpfung der orthogonalen Gruppe ist die Matrizenmultiplikation. Bei der orthogonalen Gruppe handelt es sich um eine Lie-Gruppe der Dimension .Da die Determinante einer orthogonalen Matrix nur die Werte annehmen kann, zerfällt in die beiden disjunkten Teilmengen (topologisch: Zusammenhangskomponenten) * die Drehgruppe aller Drehungen (orthogonale Matrizen mit Determinante ) und * aller Drehspiegelungen (orthogonale Matrizen mit Determinante ). (de) En matemática, el grupo ortogonal de grado n sobre un cuerpo , designado como , es el grupo de matrices ortogonales n por n con las entradas en , con la operación de grupo dada por la multiplicación de matrices. Este es un subgrupo del grupo general lineal . Cada matriz ortogonal tiene determinante 1 o -1. Las matrices n por n ortogonales con determinante 1 forman un subgrupo normal de conocido como el grupo ortogonal especial , también conocido como grupo de rotaciones. Si la característica de es 2, entonces y coinciden; en caso contrario el índice de en es 2. (es) In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. (en) En mathématiques, le groupe orthogonal est formé de transformations géométriques préservant les distances (isométries) et le point origine de l'espace. Formellement, on introduit le groupe orthogonal d'une forme quadratique q sur E, espace vectoriel sur un corps commutatif K, comme le sous-groupe du groupe linéaire GL(E) constitué des automorphismes f de E qui laissent q invariante : q(f(x)) = q(x) pour tout vecteur x de E.La loi de composition de ce groupe est la composition des applications. (fr) 数学において、n 次元の直交群（ちょっこうぐん、英: orthogonal group）とは、n 次元ユークリッド空間上のある固定された点を保つような距離を保つ変換全体からなる群であり、群の演算は変換の合成によって与える。O(n) と表記する。同値な別の定義をすれば、直交群とは、元がn×n の実直交行列であり、群の積が行列の積によって与えられるものをいう。直交行列とは、逆行列がもとの行列の転置と等しくなるような行列のことである。 直交行列の行列式は 1 か −1 である。O(n) の重要な部分群である特殊直交群 SO(n) は行列式が 1 である直交行列からなる。この群は回転群ともよばれ、例えば次元 2 や 3 では、群の元が表す変換は（2次元における）点や（3次元における）直線のまわりの通常の回転である。低次元ではこれらの群の性質は幅広く研究されている。 ここで QT は Q の転置であり、 I は単位行列である。 (ja) In de wiskunde is de orthogonale groep van over een lichaam (Ned) / veld (Be) , genoteerd als , de groep van isometrieën in de -dimensionale ruimte die de oorsprong vast houden. De orthogonale groep komt overeen met de groep van orthogonale -matrices met elementen uit men als groepsbewerking de matrixvermenigvuldiging. Dit is een ondergroep van de algemene lineaire groep bepaald door waarin de getransponeerde van is. De klassieke orthogonale groep over de reële getallen wordt meestal als geschreven. (nl) Em matemática, um grupo ortogonal é um grupo de todas as transformações lineares de um espaço vetorial de dimensões de um campo, que preserva a um não singular fixo de forma quadrática em , (ou seja, as transformações lineares tal que para todos ).Um grupo ortogonal é um . Os elementos de um grupo ortogonal são chamados transformações ortogonais de (com relação a ), ou também de automorfismos de forma . Além disso, permita (para grupos ortogonais sobre os campos com característica 2 e deixe ser a forma bilinear simétrica não singular em relacionada com o pela fórmula (pt) Ортогональна група розміру n — група перетворень евклідового простору розмірності n, які зберігають відстані. Може бути описана групою ортогональних матриць розміру n×n відносно операції множення. Позначається O(n). (uk) |
| rdfs:label | Orthogonal group (en) زمرة متعامدة (ar) Grup ortogonal (ca) Ortogonální grupa (cs) Orthogonale Gruppe (de) Grupo ortogonal (es) Groupe orthogonal (fr) Gruppo ortogonale (it) 直交群 (ja) 직교군 (ko) Orthogonale groep (nl) Grupo ortogonal (pt) Ортогональная группа (ru) Ortogonalgrupp (sv) Ортогональна група (uk) 正交群 (zh) |
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