Cartan–Dieudonné theorem

Every isometry of a finite-dimensional nondegenerate quadratic space over a field of characteristic not 2 is a product of at most as many reflections as the space’s dimension.

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Reflection (geometry): An involutive isometry that fixes a hyperplane pointwise and negates a complementary direction. These transformations are the factors whose products represent every isometry.
Nondegenerate quadratic form: A quadratic form whose associated bilinear form has zero radical. Nondegeneracy ensures the reflection construction has suitable directions available.
QR decomposition: A matrix factorization expressing a matrix as an orthogonal matrix times an upper triangular matrix. Householder reflections compute the orthogonal factor through a sequence of simple transformations.
Cartan–Dieudonné theorem over finite fields: The reflection-generation theorem specialized to nondegenerate quadratic spaces over finite fields of odd characteristic. This setting preserves the theorem while making its finite-field consequences explicit.
Orthogonal group: The group of linear transformations preserving a nondegenerate quadratic form. The theorem states that this group is generated by reflections.
Isometry: A map preserving the relevant notion of distance or quadratic form. The theorem concerns linear isometries of quadratic spaces.
Householder QR algorithm: A numerical procedure that triangularizes a matrix using Householder reflections. It operationalizes reflection-based orthogonal transformations in stable matrix computations.
Symplectic group: The group preserving a nondegenerate alternating bilinear form. Its defining form is alternating, unlike the quadratic forms covered by the theorem.
Quadratic form: A homogeneous degree-two polynomial defining a symmetric bilinear structure on a vector space. Its associated geometry determines which transformations count as isometries and reflections.
Hyperplane: A codimension-one linear subspace of a vector space. A reflection fixes a hyperplane pointwise.