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In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, is a theorem on the structure of the automorphism group of symmetric bilinear spaces.

Statement of the theorem

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.

Indefinite orthogonal group
Coordinate rotations and reflections

References

Gallier, Jean H. (2001). Geometric Methods and Applications. Texts in Applied Mathematics 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 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 67. Providence, RI: American Mathematical Society. ISBN 0-8218-1095-2. Zbl 1068.11023.

Mathematics Encyclopedia