Editing Reflection (mathematics) (section)

==Properties==

The [[matrix (mathematics)|matrix]] for a reflection is [[orthogonal matrix|orthogonal]] with [[determinant]] −1 and [[eigenvalue]]s −1, 1, 1, ..., 1. The product of two such matrices is a special orthogonal matrix that represents a rotation. Every [[Rotation (mathematics)|rotation]] is the result of reflecting in an even number of reflections in hyperplanes through the origin, and every [[improper rotation]] is the result of reflecting in an odd number. Thus reflections generate the [[orthogonal group]], and this result is known as the [[Cartan–Dieudonné theorem]].

Similarly the [[Euclidean group]], which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes.  In general, a [[group (mathematics)|group]] generated by reflections in affine hyperplanes is known as a [[reflection group]]. The [[finite group]]s generated in this way are examples of [[Coxeter group]]s.