Cartan–Dieudonné theorem

Cartan–Dieudonné theorem For other uses, see Cartan's theorem and Dieudonné's 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 (which need not be positive definite, so is not necessarily an inner product – for instance, a pseudo-Euclidean space is also a symmetric bilinear space). 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, called 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 (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 an improper rotation (3 reflections). In four dimensions, double rotations are added that represent 4 reflections.

Formal statement Let (V, b) 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 O(V, b) is a composition of at most n reflections.

See also Indefinite orthogonal group Coordinate rotations and reflections References Gallier, Jean H. (2001). Geometric Methods and Applications. Texts in Applied Mathematics. Vol. 38. Springer-Verlag. ISBN 0-387-95044-3. Zbl 1031.53001. Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004). Riemannian Geometry. Universitext. Springer-Verlag. ISBN 3-540-20493-8. Zbl 1068.53001. Garling, D. J. H. (2011). Clifford Algebras: An Introduction. London Mathematical Society Student Texts. Vol. 78. Cambridge University Press. ISBN 978-1-10742219-3. Zbl 1235.15025. Lam, T. Y. (2005). Introduction to quadratic forms over fields. Graduate Studies in Mathematics. Vol. 67. Providence, RI: American Mathematical Society. ISBN 0-8218-1095-2. Zbl 1068.11023. This abstract algebra-related article is a stub. You can help Wikipedia by expanding it.

Categories: Theorems in group theoryBilinear formsAbstract algebra stubs

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