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- Orthogonal_group abstract "In mathematics, the orthogonal group of 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. Equivalently, it is the group of n×n orthogonal matrices of a given dimension, where the group operation is given by matrix multiplication, and an orthogonal matrix is a real matrix whose inverse equals its transpose.The determinant of an orthogonal matrix being either 1 or −1, an important subgroup of O(n) is the special orthogonal group, denoted SO(n), of the orthogonal matrices of determinant 1. This group is also called the rotation group, because, 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 term "orthogonal group" may also refer to a generalization of the above case: the group of invertible linear operators that preserve a non-degenerate symmetric bilinear form or quadratic form on a vector space over a field. In particular, when the bilinear form is the scalar product on the vector space F n of dimension n over a field F, with quadratic form the sum of squares, then the corresponding orthogonal group, denoted O(n, F ), is the set of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication. This is a subgroup of the general linear group GL(n, F ) given bywhere QT is the transpose of Q and I is the identity matrix.This article mainly discusses the orthogonal groups of quadratic forms that may be expressed over some bases as the dot product; over the reals, they are the positive definite quadratic forms. Over the reals, for any non-degenerate quadratic form, there is a basis, on which the matrix of the form is a diagonal matrix such that the diagonal entries are either 1 or −1. Thus the orthogonal group depends only on the numbers of 1 and of −1, and is denoted O(p, q), where p is the number of ones and q the number of −1. For details, see indefinite orthogonal group.The derived subgroup Ω(n, F ) of O(n, F) is an often studied object because, when F is a finite field, Ω(n, F ) is often a central extension of a finite simple group.Both O(n, F ) and SO(n, F ) are algebraic groups, because the condition that a matrix be orthogonal, i.e. have its own transpose as inverse, can be expressed as a set of polynomial equations in the entries of the matrix. The Cartan–Dieudonné theorem describes the structure of the orthogonal group for a non-singular form.".
- Orthogonal_group wikiPageExternalLink ansi.altervista.org.
- Orthogonal_group wikiPageExternalLink node10.html.
- Orthogonal_group wikiPageExternalLink week105.html.
- Orthogonal_group wikiPageID "173954".
- Orthogonal_group wikiPageRevisionID "606033339".
- Orthogonal_group hasPhotoCollection Orthogonal_group.
- Orthogonal_group id "p/o070300".
- Orthogonal_group title "Orthogonal group".
- Orthogonal_group subject Category:Euclidean_symmetries.
- Orthogonal_group subject Category:Lie_groups.
- Orthogonal_group subject Category:Quadratic_forms.
- Orthogonal_group type Abstraction100002137.
- Orthogonal_group type Attribute100024264.
- Orthogonal_group type EuclideanSymmetries.
- Orthogonal_group type Form106290637.
- Orthogonal_group type Group100031264.
- Orthogonal_group type LanguageUnit106284225.
- Orthogonal_group type LieGroups.
- Orthogonal_group type Part113809207.
- Orthogonal_group type Property104916342.
- Orthogonal_group type QuadraticForms.
- Orthogonal_group type Relation100031921.
- Orthogonal_group type SpatialProperty105062748.
- Orthogonal_group type Symmetry105064827.
- Orthogonal_group type Word106286395.
- Orthogonal_group comment "In mathematics, the orthogonal group of 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.".
- Orthogonal_group label "Groupe orthogonal".
- Orthogonal_group label "Grupo ortogonal".
- Orthogonal_group label "Gruppo ortogonale".
- Orthogonal_group label "Orthogonal group".
- Orthogonal_group label "Orthogonale Gruppe".
- Orthogonal_group label "Orthogonale groep".
- Orthogonal_group label "Ортогональная группа".
- Orthogonal_group label "زمرة متعامدة".
- Orthogonal_group label "正交群".
- Orthogonal_group sameAs Ortogonální_grupa.
- Orthogonal_group sameAs Orthogonale_Gruppe.
- Orthogonal_group sameAs Grupo_ortogonal.
- Orthogonal_group sameAs Groupe_orthogonal.
- Orthogonal_group sameAs Gruppo_ortogonale.
- Orthogonal_group sameAs 직교군.
- Orthogonal_group sameAs Orthogonale_groep.
- Orthogonal_group sameAs m.017gp1.
- Orthogonal_group sameAs Q1783179.
- Orthogonal_group sameAs Q1783179.
- Orthogonal_group sameAs Orthogonal_group.
- Orthogonal_group wasDerivedFrom Orthogonal_group?oldid=606033339.
- Orthogonal_group isPrimaryTopicOf Orthogonal_group.