In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. This is a subgroup of the general linear group GL(n,F).

Every orthogonal matrix has determinant either 1 or -1. The orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O(n,F) known as the special orthogonal group SO(n,F). If the characteristic of F is 2, then O(n,F) and SO(n,F) coincide; otherwise the index of SO(n,F) in O(n,F) is 2.

Both O(n,F) and SO(n,F) 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.

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 connected component containing the identity matrix.

Both the real orthogonal and real special orthogonal groups have simple geometric interpretations. O(n,R) is isomorphic to the group of isometries of Rⁿ which leave the origin fixed. SO(n,R) is isomorphic to the group of rotations of Rⁿ that keep the origin fixed.

SO(2,R) is isomorphic (as a Lie group) to the circle S¹, consisting of all complex numbers of absolute value 1, with multiplication of complex numbers as group operation. This isomorphism sends the complex number exp(φi) = cos(φ) + i sin(φ) to the orthogonal matrix

<math>\begin{bmatrix}\cos(\phi)&-\sin(\phi)\\

\sin(\phi)&\cos(\phi)\end{bmatrix}<math>

The group SO(3,R), understood as the set of rotations of 3-dimensional space, is of major importance in the sciences and engineering. For a detailed description, see rotation group.

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 Lie algebra associated to the Lie groups O(n,R) and SO(n,R) consists of the skew-symmetric real n-by-n matrices, with the Lie bracket given by the commutator. This Lie algebra is often denoted by o(n,R) or by so(n,R).

Over the complex number field

Over the field C of complex numbers, O(n,C) and SO(n,C) are complex Lie groups of dimension n(n-1)/2 over C (which means the dimension over R is twice that). O(n,C) has two connected components, and SO(n,C) is the connected component containing the identity matrix. For n ≥ 2 these groups are noncompact.

Just as in the real case SO(n,C) is not simply connected. For n > 2 the fundamental group of SO(n,C) is cyclic of order 2 whereas the fundamental group of SO(2,C) is infinite cyclic.

The complex Lie algebra associated to O(n,C) and SO(n,C) consists of the skew-symmetric complex n-by-n matrices, with the Lie bracket given by the commutator.

Related topics

• rotation group, SO(3,R)
• generalized orthogonal group
• unitary group
• symplectic group

de:Orthogonale Gruppe