Orthogonal group

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 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) 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.

The topology of the real special orthogonal groups may be described this way:

SO(2,R) is isomorphic to the circle S¹, consisting of all complex numbers of absolute value 1, with multiplication of complex numbers as group operation.

For n≥2, SO(n,R) is not simply connected and the topology is harder to visualize. For the case n=3, consider the solid ball in R³ of radius π (that is, all points of R³ of distance π or less from the origin). For every point in this ball there is a rotation, with axis through the point and rotation angle equal to the distance of the point from the origin. Rotation through angles between 0 and -π correspond to the point on the same axis and distance from the origin but on the opposite side of the origin. The one remaining issue is that the two rotations through π and through -π are the same. So we identify ("glue together") antipodal points on the surface of the ball. This identification illustrates the non-simply-connected nature of the topology. (See charts on SO(3) for further details.)

In terms of algebraic topology, for SO(n,R) the first homotopy group is Z₂, and the spinor group Spin(n) is its universal cover.

The Lie algebra associated to 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.

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. They are not compact if n≥2. O(n,C) has two connected components, and SO(n,C) is the connected component containing the identity matrix. SO(n,C) is simply connected. The Lie algebra associated to O(n,R) and SO(n,R) consists of the skew-symmetric complex n-by-n matrices, with the Lie bracket given by the commutator.