Orthogonal Group - Over The Real Number Field

Over The Real Number Field

Over the field R of real numbers, the orthogonal group O(n, R) and the special orthogonal group SO(n, R) are often simply denoted by O(n) and SO(n) if no confusion is possible. They form real compact Lie groups of dimension n(n − 1)/2. O(n, R) has two connected components, with SO(n, R) being the identity component, i.e., the connected component containing the identity matrix.

The real orthogonal and real special orthogonal groups have the following geometric interpretations:

O(n, R) is a subgroup of the Euclidean group E(n), the group of isometries of Rn; it contains those that leave the origin fixed – O(n, R) = E(n) ∩ GL(n, R). It is the symmetry group of the sphere (n = 3) or hypersphere and all objects with spherical symmetry, if the origin is chosen at the center.

SO(n, R) is a subgroup of E+(n), which consists of direct isometries, i.e., isometries preserving orientation; it contains those that leave the origin fixed – SO(n, R) = E+(n) ∩ GL(n, R) = E(n) ∩ GL+(n, R). It is the rotation group of the sphere and all objects with spherical symmetry, if the origin is chosen at the center.

{±I} is a normal subgroup and even a characteristic subgroup of O(n, R), and, if n is even, also of SO(n, R). If n is odd, O(n, R) is the internal direct product of SO(n, R) and {±I}. The cyclic group of k-fold rotations C_k is for every positive integer k a normal subgroup of O(2, R) and SO(2, R).

Relative to suitable orthogonal bases, the isometries are of the form:

where the matrices R₁, ..., R_k are 2-by-2 rotation matrices in orthogonal planes of rotation. As a special case, known as Euler's rotation theorem, any (non-identity) element of SO(3, R) is rotation about a uniquely defined axis.

The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter group, and elements have length at most n (require at most n reflections to generate; this follows from the above classification, noting that a rotation is generated by 2 reflections, and is true more generally for indefinite orthogonal groups, by the Cartan–Dieudonné theorem). A longest element (element needing the most reflections) is reflection through the origin (the map v ↦ −v), though so are other maximal combinations of rotations (and a reflection, in odd dimension).

The symmetry group of a circle is O(2, R), also called Dih(S1), where S1 denotes the multiplicative group of complex numbers of absolute value 1.

SO(2, R) is isomorphic (as a real Lie group) to S1. This isomorphism sends the complex number exp(φi) = cos(φ) + i sin(φ) to the orthogonal matrix

The group SO(3, R), understood as the set of rotations of 3-dimensional space, is of major importance in the sciences and engineering, and there are numerous charts on SO(3).

In terms of algebraic topology, for n > 2 the fundamental group of SO(n, R) is cyclic of order 2, and the spinor group Spin(n) is its universal cover. For n = 2 the fundamental group is infinite cyclic and the universal cover corresponds to the real line (the spinor group Spin(2) is the unique 2-fold cover).