Orthogonal Group

The orthogonal group , for every dimension , is the group of orthogonal matrices. These matrices form a group because they are closed under matrix multiplication and taking matrix inverses.

Thinking of a matrix as given by coordinate functions, the set of matrices is identified with . Writing an matrix as , the orthogonal matrices are the solutions to the equations

(1)

where is the identity matrix, which are redundant. Only of these are independent, leaving "free variables." In fact, the orthogonal group is a smooth -dimensional submanifold.

Because the orthogonal group is a group and a manifold, it is a Lie group. has a submanifold tangent space at the identity element that is the Lie algebra of antisymmetric matrices . In fact, the orthogonal group is a compact Lie group.

The determinant of an orthogonal matrix is either 1 or , and so the orthogonal group has two connected components. The connected component containing the identity element is the special orthogonal group . For example, the group has group action on the plane by rotations and reflections:

(2)

where is any real number in . These matrices preserve the quadratic form , and so they also preserve circles , which are the group orbits.

As a manifold, consists of two disjoint copies of the circle.

There are several generalizations of the orthogonal group. First, it is possible to define the orthogonal group for any nondegenerate real symmetric bilinear form with matrix signature . The group of matrices which preserve , that is,

(3)

for all vectors and , is denoted . The Lorentz group is . For example, the matrices

(4)

are elements of . They preserve the quadratic form so they preserve the hyperbolas .

Instead of using real numbers for the coefficients, it is possible to use coefficients from any field of field characteristic different from 2, in which case it is denoted . The orthogonal matrices still satisfy . For example, contains

(5)

and has 48 elements in total.

More generally, for a specified nonsingular quadratic form over a field , the orthogonal group consists of the invertible matrices satisfying for all vectors . The notation uses the matrix signature of a real form, not a classification over arbitrary fields. In field characteristic 2, preservation of the associated symmetric bilinear form need not imply preservation of . Orthogonal groups over finite fields belong to the classical families of Lie-type groups.

When the coefficients are complex numbers, it is called the complex orthogonal group, which is much different from the unitary group. For example, matrices of the form

(6)

are in . In particular, is a complex Lie group that is not a compact Lie group for . The exceptional group is finite and compact. The equations defining in affine space are polynomials of degree two. Consequently, is a linear algebraic group.

The numbers of conjugacy classes of finite subgroups of orders , 2, 3, ... in the orthogonal group are 1, 3, 1, 5, 1, 5, 1, 7, 1, 5, 1, 8, ... (OEIS A001051), i.e., a repeating sequence of copies of with the exceptions , , , , and .