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 .
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:
- .
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:
- 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:
- Shift of by the support vector of the starting line
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 .