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 (the origin), 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, since its elements are the rotations around the origin. In low dimension, these groups have been widely studied, see , and . The other component consists of all orthogonal matrices of determinant −1. 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 formFor base fields of characteristic not 2, the definition in terms of a symmetric bilinear form is equivalent to that in terms of a quadratic form, but in characteristic 2 these notions differ. 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 (homothety), the linear maps from to that map orthogonal vectors to orthogonal vectors.
In Euclidean geometry
The orthogonal 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.
is a normal subgroup of , as being the kernel of the determinant, which is a group homomorphism whose image is the multiplicative group . This implies that the orthogonal group is an internal semidirect product of and any subgroup formed with the identity and a reflection.
The group with two elements (where is the identity matrix) is a normal subgroup and even a characteristic subgroup of , and, if is even, also of . If is odd, is the internal direct product of and .
The group is abelian (whereas is not abelian when ). Its finite subgroups are the cyclic group of -fold rotations, for every positive integer . All these groups are normal subgroups of and .
Canonical form For any element of there is an orthogonal basis, where its matrix has the formF. Gantmacher, Theory of matrices, vol. 1, Chelsea, 1959, p. 285.Serge Lang, Linear Algebra, 3rd ed., Springer, 1987, p. 230.
where there may be any number, including zero, of ±1's; and where the matrices are 2-by-2 rotation matrices, that is matrices of the form
with .
This results from the spectral theorem by regrouping eigenvalues that are complex conjugate, and taking into account that the absolute values of the eigenvalues of an orthogonal matrix are all equal to .
The element belongs to if and only if there are an even number of on the diagonal. A pair of eigenvalues can be identified with a rotation by and a pair of eigenvalues can be identified with a rotation by .
The special case of is known as Euler's rotation theorem, which asserts that every (non-identity) element of is a rotation about a unique axis–angle pair.
Reflections Reflections are the elements of whose canonical form is
where is the identity matrix, and the zeros denote row or column zero matrices. In other words, a reflection is a transformation that transforms the space in its mirror image with respect to a hyperplane.
In dimension two, every rotation can be decomposed into a product of two reflections. More precisely, a rotation of angle is the product of two reflections whose axes form an angle of .
A product of up to elementary reflections always suffices to generate any element of . This results immediately from the above canonical form and the case of dimension two.
The Cartan–Dieudonné theorem is the generalization of this result to the orthogonal group of a nondegenerate quadratic form over a field of characteristic different from two.
The reflection through the origin (the map ) is an example of an element of that is not a product of fewer than reflections.
Symmetry group of spheres The orthogonal group is the symmetry group of the -sphere (for , this is just the sphere) and all objects with spherical symmetry, if the origin is chosen at the center.
The symmetry group of a circle is . The orientation-preserving subgroup is isomorphic (as a real Lie group) to the circle group, also known as , the multiplicative group of the complex numbers of absolute value equal to one. This isomorphism sends the complex number of absolute value to the special orthogonal matrix
In higher dimension, has a more complicated structure (in particular, is no longer commutative). The topological structures of the -sphere and are strongly correlated, and this correlation is widely used for studying both topological spaces.
Group structure The groups and are real compact Lie groups of dimension . The group has two connected components, with being the identity component, that is, the connected component containing the identity matrix.
As algebraic groups The orthogonal group can be identified with the group of the matrices such that . Since both members of this equation are symmetric matrices, this provides equations that the entries of an orthogonal matrix must satisfy, and which are not all satisfied by the entries of any non-orthogonal matrix.
This proves that is an algebraic set. Moreover, it can be proved that its dimension is
which implies that is a complete intersection. This implies that all its irreducible components have the same dimension, and that it has no embedded component. In fact, has two irreducible components, that are distinguished by the sign of the determinant (that is or ). Both are nonsingular algebraic varieties of the same dimension . The component with is .
Maximal tori and Weyl groups A maximal torus in a compact Lie group G is a maximal subgroup among those that are isomorphic to for some , where is the standard one-dimensional torus. Theorem 11.2
In and , for every maximal torus, there is a basis on which the torus consists of the block-diagonal matrices of the form
where each belongs to . In and , the maximal tori have the same form, bordered by a row and a column of zeros, and on the diagonal.
The Weyl group of is the semidirect product of a normal elementary abelian 2-subgroup and a symmetric group, where the nontrivial element of each factor of acts on the corresponding circle factor of } by inversion, and the symmetric group acts on both and } by permuting factors. The elements of the Weyl group are represented by matrices in . The factor is represented by block permutation matrices with 2-by-2 blocks, and a final on the diagonal. The component is represented by block-diagonal matrices with 2-by-2 blocks either
with the last component chosen to make the determinant .
The Weyl group of is the subgroup of that of , where is the kernel of the product homomorphism given by ; that is, is the subgroup with an even number of minus signs. The Weyl group of is represented in by the preimages under the standard injection of the representatives for the Weyl group of . Those matrices with an odd number of blocks have no remaining final coordinate to make their determinants positive, and hence cannot be represented in .
Topology
Low-dimensional topology
The low-dimensional (real) orthogonal groups are familiar spaces: , a two-point discrete space, consisting of the identity map, and the reflection map on the line. , since only the identity has determinant 1. is , as explained in the section "Symmetry group of spheres" above. has the structure of real projective space, . Section 1.3.4 is doubly covered by , as explained below.
Fundamental group In terms of algebraic topology, for the fundamental group of is cyclic of order 2, Proposition 13.10 and the spin group is its universal cover. For the fundamental group is infinite cyclic and the universal cover corresponds to the real line (the group is the unique connected 2-fold cover).
Homotopy groups Generally, the homotopy groups of the real orthogonal group are related to homotopy groups of spheres, and thus are in general hard to compute. However, one can compute the homotopy groups of the stable orthogonal group (aka the infinite orthogonal group), defined as the direct limit of the sequence of inclusions:
Since the inclusions are all closed, hence cofibrations, this can also be interpreted as a union. On the other hand, is a homogeneous space for , and one has the following fiber bundle: