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 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.

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 (which can be written as the composition of two reflections).

In four dimensions, double rotations are added that represent 4 reflections.

Let (V, b) be an n-dimensional, non-degenerate symmetric bilinear space over a field with characteristic not equal to 2.