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 notion of a symmetric bilinear space is a generalization of
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
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
References
- Zbl 1031.53001.
- Gallot, Sylvestre; Zbl 1068.53001.
- Garling, D. J. H. (2011). Clifford Algebras: An Introduction. London Mathematical Society Student Texts. Vol. 78. Zbl 1235.15025.
- Lam, T. Y. (2005). Introduction to quadratic forms over fields. Zbl 1068.11023.
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