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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. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n × n orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact.

The orthogonal group in dimension n has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO(n). It consists of all orthogonal matrices of determinant 1. This group is also called the rotation group, generalizing the fact that 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 other component consists of all orthogonal matrices of determinant −1. This component does not form a group, as the product of any two of its elements is of determinant 1, and therefore not an element of the component.

By extension, for any field F, an n × n matrix with entries in F such that its inverse equals its transpose is called an orthogonal matrix over F. The n × n orthogonal matrices form a subgroup, denoted O(n, F), of the general linear group GL(n, F); that is

$${\displaystyle \operatorname {O} (n,F)=\left\{Q\in \operatorname {GL} (n,F)\mid Q^{\mathsf {T}}Q=QQ^{\mathsf {T}}=I\right\}.}$$

More generally, given a non-degenerate symmetric bilinear form or quadratic form^[1] on a vector space over a field, the orthogonal group of the form is the group of invertible linear maps that preserve the form. The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the dot product, or, equivalently, the quadratic form is the sum of the square of the coordinates.

All orthogonal groups are algebraic groups, since the condition of preserving a form can be expressed as an equality of matrices.

Name

The name of "orthogonal group" originates from the following characterization of its elements. Given a Euclidean vector space E of dimension n, the elements of the orthogonal group O(n) are, up to a uniform scaling (homothecy), the linear maps from E to E that map orthogonal vectors to orthogonal vectors.

In Euclidean geometry

The orthogonal O(n) is the subgroup of the general linear group GL(n, R), consisting of all endomorphisms that preserve the Euclidean norm; that is, endomorphisms g such that ${\displaystyle \|g(x)\|=\|x\|.}$

Let E(n) be the group of the Euclidean isometries of a Euclidean space S of dimension n. This group does not depend on the choice of a particular space, since all Euclidean spaces of the same dimension are isomorphic. The stabilizer subgroup of a point x ∈ S is the subgroup of the elements g ∈ E(n) such that g(x) = x. This stabilizer is (or, more exactly, is isomorphic to) O(n), since the choice of a point as an origin induces an isomorphism between the Euclidean space and its associated Euclidean vector space.

There is a natural group homomorphism p from E(n) to O(n), which is defined by

${\displaystyle p(g)(y-x)=g(y)-g(x),}$

where, as usual, the subtraction of two points denotes the translation vector that maps the second point to the first one. This is a well defined homomorphism, since a straightforward verification shows that, if two pairs of points have the same difference, the same is true for their images by g (for details, see Affine space § Subtraction and Weyl's axioms).

The kernel of p is the vector space of the translations. So, the translations form a normal subgroup of E(n), the stabilizers of two points are conjugate under the action of the translations, and all stabilizers are isomorphic to O(n).

Moreover, the Euclidean group is a semidirect product of O(n) and the group of translations. It follows that the study of the Euclidean group is essentially reduced to the study of O(n).

Special orthogonal group

By choosing an orthonormal basis of a Euclidean vector space, the orthogonal group can be identified with the group (under matrix multiplication) of orthogonal matrices, which are the matrices such that

${\displaystyle QQ^{\mathsf {T}}=I.}$

It follows from this equation that the square of the determinant of Q equals 1, and thus the determinant of Q is either 1 or −1. The orthogonal matrices with determinant 1 form a subgroup called the special orthogonal group, denoted SO(n), consisting of all direct isometries of O(n), which are those that preserve the orientation of the space.

SO(n) is a normal subgroup of O(n), as being the kernel of the determinant, which is a group homomorphism whose image is the multiplicative group {−1, +1}. This implies that the orthogonal group is an internal semidirect product of SO(n) and any subgroup formed with the identity and a reflection.

The group with two elements {±I} (where I is the identity matrix) is a normal subgroup and even a characteristic subgroup of O(n), and, if n is even, also of SO(n). If n is odd, O(n) is the internal direct product of SO(n) and {±I}.

The group SO(2) is abelian (this is not the case of SO(n) for every n > 2). Its finite subgroups are the cyclic group C_k of k-fold rotations, for every positive integer k. All these groups are normal subgroups of O(2) and SO(2).

Canonical form

For any element of O(n) there is an orthogonal basis, where its matrix has the form

${\displaystyle {\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}},}$

where the matrices R₁, ..., R_k are 2-by-2 rotation matrices, that is matrices of the form