Cartan–Dieudonné theorem
In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an n-dimensional symmetric bilinear space can be described as the composition of at most n reflections.
The notion of a symmetric bilinear space is a generalization of Euclidean space whose structure is defined by a symmetric bilinear form. The orthogonal transformations in the space are those automorphisms which preserve the value of the bilinear form between every pair of vectors; in Euclidean space, this corresponds to preserving distances and angles. These orthogonal transformations form a group under composition, the orthogonal group.
For example, in the two dimensional Euclidean plane, every orthogonal transformation is either a reflection across a line through the origin or a rotation about the origin. Any arbitrary composition of such rotations and reflections can be rewritten as a composition of no more than 2 reflections. Similarly, in three-dimensional Euclidean space, every orthogonal transformation can be described as a single reflection, a rotation, or an improper rotation. In four-dimensions, double rotations are added that represent 4 reflections.Formal statement
Let be an n-dimensional, non-degenerate symmetric bilinear space over a field with characteristic not equal to 2. Then, every element of the orthogonal group is a composition of at most n reflections.
