In mathematics, the orthogonal group of a symmetric bilinear form or quadratic form on a vector space is the group of invertible linear operators on the space which preserve the form: it is a subgroup of the automorphism group of the vector space. The Cartan–Dieudonné theorem describes the structure of the orthogonal group for a non-singular form.
In particular, when the bilinear form is the scalar product on the vector space Fn of dimension n over a field F, with quadratic form the sum of squares, then the corresponding orthogonal group, written as 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 by
where QT is the transpose of Q and I is the identity matrix. The classical orthogonal group over the real numbers is usually just written O(n).
This article mainly discusses definite forms: the orthogonal group of the positive definite form (equivalent to the sum of n squares). Negative definite forms (equivalent to the negative sum of n squares) are identical since O(n, 0) = O(0, n). However, the associated Pin groups differ.
Every orthogonal matrix has determinant either 1 or −1. The orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O(n, F) known as the special orthogonal group, SO(n, F). (More precisely, SO(n, F) is the kernel of the Dickson invariant, discussed below.) By analogy with GL/SL (general linear group, special linear group), the orthogonal group is sometimes called the general orthogonal group and denoted GO, though this term is also sometimes used for indefinite orthogonal groups O(p, q).
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.
Read more about Orthogonal Group: Over The Real Number Field, Over The Complex Number Field, Topology, Over Finite Fields, The Dickson Invariant, Orthogonal Groups of Characteristic 2, The Spinor Norm, Galois Cohomology and Orthogonal Groups, Related Groups, Principal Homogeneous Space: Stiefel Manifold
Other articles related to "orthogonal group, orthogonal, group":
... The principal homogeneous space for the orthogonal group O(n) is the Stiefel manifold Vn(Rn) of orthonormal bases (orthonormal n-frames) ... In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point given an orthogonal space, there is no natural choice of orthonormal basis, but ... map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis ...
... In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space ... It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group ... In general the map from the Pin group to the orthogonal group is not onto or a universal covering space, but if the quadratic form is definite (and dimension is greater ...
... As with the orthogonal group, the projective orthogonal group can be generalized in two main ways changing the field or changing the quadratic form ... The complex projective orthogonal group, PO(n,C) should not be confused with the projective unitary group, PU(n) PO preserves a symmetric form, while PU preserves a ...
... genus as f, and Aut(Λ) is the automorphism group of Λ ... mod p r such that where A is the Gram matrix of f, or in other words the order of the automorphism group of the form reduced mod p r ... The factor of 2 in front represents the Tamagawa number of the special orthogonal group, which is only 1 in dimensions 0 and 1 ...
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“We begin with friendships, and all our youth is a reconnoitering and recruiting of the holy fraternity they shall combine for the salvation of men. But so the remoter stars seem a nebula of united light, yet there is no group which a telescope will not resolve; and the dearest friends are separated by impassable gulfs.”
—Ralph Waldo Emerson (18031882)