Reflection matrix

As a reflection matrix is known in linear algebra , a matrix , a reflection is. The simplest example is the reflection on a straight line through the origin in the plane with the angle of inclination . The reflection mapping results as the matrix-vector product of the matrix with the corresponding vector . ${\displaystyle g}$ ${\displaystyle \alpha }$

Reflection on a plane straight line through the origin

The matrix of a reflection on a straight line through the origin with the angle to the positive x-axis is: ${\displaystyle S_{g}}$ ${\displaystyle \alpha }$

${\displaystyle S_{g}={\begin{pmatrix}\cos 2\alpha &\sin 2\alpha \\\sin 2\alpha &-\cos 2\alpha \end{pmatrix}}}$.

Reflection on any straight line

This also enables a representation of the reflection of a vector on any straight line with an angle of inclination . To do this, two steps have to be carried out: ${\displaystyle {\vec {v}}}$ ${\displaystyle g={\vec {a}}+r\cdot {\vec {u}}}$ ${\displaystyle \alpha }$

1. It is attributed to a reflection on a straight line through the origin . This is by shifting to achieved . The vector is now mirrored to: ${\displaystyle g^{*}=r\cdot {\vec {u}}}$ ${\displaystyle g}$ ${\displaystyle -{\vec {a}}}$ ${\displaystyle {\vec {v}}'={\vec {v}}-{\vec {a}}}$ ${\displaystyle {\vec {v}}'}$ ${\displaystyle g^{*}}$

   ${\displaystyle {\vec {q}}'=S_{g}({\vec {v}}')=S_{g}({\vec {v}}-{\vec {a}})={\begin{pmatrix}\cos 2\alpha &\sin 2\alpha \\\sin 2\alpha &-\cos 2\alpha \end{pmatrix}}\cdot ({\vec {v}}-{\vec {a}})}$

2. Shift of by the support vector of the starting line ${\displaystyle {\vec {q}}'}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle g}$

   ${\displaystyle {\vec {q}}={\vec {q}}'+{\vec {a}}={\begin{pmatrix}\cos 2\alpha &\sin 2\alpha \\\sin 2\alpha &-\cos 2\alpha \end{pmatrix}}\cdot ({\vec {v}}-{\vec {a}})+{\vec {a}}}$

More general reflections

Reflection matrices are orthogonal matrices and have the determinant −1.

The representations of reflections on hyperplanes are called Householder matrices in numerical mathematics .

literature

• Wolfgang Mackens, Heinrich Voss: Mathematics. For engineering students. Volume 1. HECO-Verlag, Aachen 1993, ISBN 3-930121-00-X .