Inmathematics, theCartan–Dieudonné theorem, named afterÉlie Cartan andJean Dieudonné, establishes that everyorthogonal transformation in ann-dimensional symmetric bilinear space can be described as thecomposition of at mostnreflections.
The notion of a symmetric bilinear space is a generalization ofEuclidean space whose structure is defined by asymmetric bilinear form (which need not bepositive definite, so is not necessarily aninner product – for instance, apseudo-Euclidean space is also a symmetric bilinear space). The orthogonal transformations in the space are thoseautomorphisms which preserve the value of the bilinear form between every pair of vectors; in Euclidean space, this corresponds to preserving distances andangles. These orthogonal transformations form agroup under composition, called theorthogonal group.
For example, in the two-dimensionalEuclidean plane, every orthogonal transformation is either a reflection across aline through the origin or arotation about the origin (which can be written as the composition of two reflections). 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 (2 reflections), or animproper rotation (3 reflections). In four dimensions,double rotations are added that represent 4 reflections.
Let(V,b) be ann-dimensional,non-degenerate symmetric bilinear space over afield withcharacteristic not equal to 2. Then, every element of the orthogonal groupO(V,b) is a composition of at mostn reflections.
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