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. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact.
The orthogonal group in dimension has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted . It consists of all orthogonal matrices of determinant 1. This group is also called the rotation group, generalizing the fact that 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 other component consists of all orthogonal matrices of determinant . This component does not form a group, as the product of any two of its elements is of determinant 1, and therefore not an element of the component.
By extension, for any field , an matrix with entries in such that its inverse equals its transpose is called an orthogonal matrix over . The orthogonal matrices form a subgroup, denoted , of the general linear group ; that is
More generally, given a non-degenerate symmetric bilinear form or quadratic form on a vector space over a field, the orthogonal group of the form is the group of invertible linear maps that preserve the form. The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the dot product, or, equivalently, the quadratic form is the sum of the square of the coordinates.
All orthogonal groups are algebraic groups, since the condition of preserving a form can be expressed as an equality of matrices.
Name
The name of "orthogonal group" originates from the following characterization of its elements. Given a Euclidean vector space of dimension , the elements of the orthogonal group are, up to a uniform scaling (homothecy), the linear maps from to that map orthogonal vectors to orthogonal vectors.
In Euclidean geometry
The orthogonal group is the subgroup of the general linear group , consisting of all endomorphisms that preserve the Euclidean norm; that is, endomorphisms such that
Let be the group of the Euclidean isometries of a Euclidean space of dimension . This group does not depend on the choice of a particular space, since all Euclidean spaces of the same dimension are isomorphic. The stabilizer subgroup of a point is the subgroup of the elements such that . This stabilizer is (or, more exactly, is isomorphic to) , since the choice of a point as an origin induces an isomorphism between the Euclidean space and its associated Euclidean vector space.
There is a natural group homomorphism from to , which is defined by
where, as usual, the subtraction of two points denotes the translation vector that maps the second point to the first one. This is a well defined homomorphism, since a straightforward verification shows that, if two pairs of points have the same difference, the same is true for their images by (for details, see ).
The kernel of is the vector space of the translations. So, the translations form a normal subgroup of , the stabilizers of two points are conjugate under the action of the translations, and all stabilizers are isomorphic to .
Moreover, the Euclidean group is a semidirect product of and the group of translations. It follows that the study of the Euclidean group is essentially reduced to the study of .
Special orthogonal group
By choosing an orthonormal basis of a Euclidean vector space, the orthogonal group can be identified with the group (under matrix multiplication) of orthogonal matrices, which are the matrices such that
It follows from this equation that the square of the determinant of equals , and thus the determinant of is either or . The orthogonal matrices with determinant form a subgroup called the special orthogonal group, denoted , consisting of all direct isometries of , which are those that preserve the orientation of the space.