|
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. Equivalently, it is the group of orthogonal matrices, 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 or , an important subgroup of is the special orthogonal group, denoted SO(''n''), of the orthogonal matrices of determinant . 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 , and . 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〔For base fields of characteristic not 2, it is equivalent to use symmetric bilinear forms or quadratic forms. But in characteristic 2 these notions differ.〕 on a vector space over a field. In particular, when the bilinear form is the scalar product on the vector space of dimension over a field , with quadratic form the sum of squares, then the corresponding orthogonal group, denoted , is the set of orthogonal matrices with entries from , with the group operation of matrix multiplication. This is a subgroup of the general linear group given by : where T is the transpose of and 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 or . Thus the orthogonal group depends only on the numbers of and of , and is denoted , where is the number of ones and the number of negative ones. For details, see indefinite orthogonal group. The derived subgroup of is an often studied object because, when is a finite field, is often a central extension of a finite simple group. Both and 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. ==Name== The determinant of any orthogonal matrix is either or . The orthogonal -by- matrices with determinant form a normal subgroup of known as the special orthogonal group , consisting of all proper rotations. (More precisely, 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 , though this term is also sometimes used for ''indefinite'' orthogonal groups . The term rotation group can be used to describe either the special or general orthogonal group. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Orthogonal group」の詳細全文を読む スポンサード リンク
|