Orthogonal group

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. 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. 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. 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. 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 by where 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 negative ones. 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. The determinant of any orthogonal matrix is 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 ), consisting of all proper rotations. (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 term rotation group can be used to describe either the special or general orthogonal group. The structure of the orthogonal group differs in certain respects between even and odd dimensions; for example, over ordered fields (such as R) the −I element is orientation-preserving in even dimensions, but orientation-reversing in odd dimensions. When this distinction is to be emphasized, the groups may be denoted O(2k) and O(2k + 1), reserving n for the dimension of the space (n = 2k or n = 2k + 1). The letters p or r are also used, indicating the rank of the corresponding Lie algebra; in odd dimension the corresponding Lie algebra is s o ( 2 r + 1 ) {displaystyle {mathfrak {so}}(2r+1)} , while in even dimension the Lie algebra is s o ( 2 r ) {displaystyle {mathfrak {so}}(2r)} .