Orthogonal Group

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.

For other non-singular forms O(p,q), see indefinite orthogonal group.

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.