The Cartan-Dieudonne Theorem establishes that every orthogonal transformation of an n-dimensional symmetric bilinear space can be written as a composition of at most n reflections. Named for Elie Cartan and Jean Dieudonne, it applies to any space whose structure is defined by a symmetric bilinear form, including ordinary Euclidean space, where the theorem says every distance- and angle-preserving transformation is built from reflections alone; in the Euclidean plane, for instance, every orthogonal transformation is either a single reflection or a rotation formed from two reflections.
Facts
StatementEvery orthogonal transformation of an n-dimensional non-degenerate symmetric bilinear space over a field of characteristic not equal to 2 is a composition of at most n reflections. 2 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Proved By
Source Cartan-Dieudonne theorem (Wikipedia)
Sources
1. Wikipedia: Cartan-Dieudonné theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
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.
2. Cartan-Dieudonne theorem (Wikipedia)
Formal statement
every element of the orthogonal group O(V, b) is a composition of at most n reflections
Proved By: Elie Cartan, Lead paragraph
In mathematics, the Cartan-Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an n-dimensional
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