Orthogonal Group

(group theory) For given n and field F (especially where F is the real numbers), the group of n × n orthogonal matrices with elements in F, where the group operation is matrix multiplication.

Denoted O(n) in the real number case; O(n, F) in the general case.

In the case that F is the real numbers, the orthogonal group is equivalently definable as the group of distance-preserving transformations of an n-dimensional Euclidean space that preserve a given fixed point, where the group operation is that of composition of transformations.