Reflections Generate the Orthogonal Group
Every element of is a product of at most reflections.
Theorem 9.14 (Reflections Generate )
Every is a product of at most reflections.
Proof
Use induction on . For , and the matrix is a reflection. For the inductive step, set . Then fixes and preserves the hyperplane .
By induction, its restriction to this hyperplane is a product of at most reflections. Both sides also fix , so the equality holds on . Multiplying by gives a product of at most reflections for .
Related
Stated in
- Theorem 9.14 (Reflections Generate )§9.3 Orthogonal Groups