Cartan–Dieudonné theorem

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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.

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.

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