Orthogonal group

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. The elements of SO(n,R) can be interpreted as the rotations in Rⁿ that keep the origin fixed.

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. SO(n,R) is not simply connected for n≥2; 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.