Orthogonal group

In mathematics, the orthogonal group in dimension $n$ , denoted $O(n)$ , is the group of distance-preserving transformations of a Euclidean space of dimension $n$ that preserve a fixed point, where the group operation is given by composing transformations. Equivalently, it is the group of $n \times n$ orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix is a real matrix whose inverse equals its transpose.

An important subgroup of $O(n)$ is the special orthogonal group, denoted $SO(n)$ , of the orthogonal matrices of determinant $1$ . This group is also called the rotation group, because, in dimensions 2 and 3, its elements are the usual rotations around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see $SO(2)$ , $SO(3)$ and $SO(4)$ .

The term "orthogonal group" may also refer to a generalization of the above case: the group of invertible linear operators that preserve a non-degenerate symmetric bilinear form or quadratic form on a vector space over a field. In particular, when the bilinear form is the scalar product on the vector space $F n$ of dimension $n$ over a field $F$ , with quadratic form the sum of squares, then the corresponding orthogonal group, denoted $O(n, F)$ , is the set of $n \times n$ orthogonal matrices with entries from $F$ , with the group operation of matrix multiplication. This is a subgroup of the general linear group $GL(n, F)$ given by

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