75. Reflection Matrix
A reflection matrix maps each point to its mirror image across a fixed line (in 2D), plane (in 3D), or hyperplane (in ).
75.1. Reflection across a line through the origin (2D)
Line making angle with the -axis:
Special cases:
| Axis | Matrix |
|---|---|
| -axis () | |
| -axis () | |
| Line () | |
| Line () |
75.2. Reflection across a hyperplane (general)
For a hyperplane through the origin with unit normal , the reflection is the Householder matrix:
This subtracts twice the projection onto from each input.
Example
Reflect across the plane in . Normal: .
flips sign, and preserved. ✓
75.3. Properties
- Orthogonal: , so (reflections are self-inverse — reflecting twice is the identity)
- Determinant: — orientation-reversing
- Eigenvalues: (for the direction perpendicular to the mirror) and with multiplicity (for directions parallel to the mirror)
- Preserves lengths and angles (it’s orthogonal)
- Involution:
75.4. Rotations vs reflections
| Rotation | Reflection | |
|---|---|---|
| Orientation | preserved | reversed |
| Self-inverse? | no (unless or ) | yes |
| Symmetric? | no | yes |
Composition: two reflections produce a rotation (by , where is the angle between the mirrors).
75.5. Householder transformations
Reflections of the form (Householder transformations) are the building blocks of stable numerical algorithms for:
- QR decomposition — alternative to Gram–Schmidt, much more stable
- Tridiagonalization of symmetric matrices (preprocessing step for eigenvalue computation)
75.6. See also
- Rotation Matrix
- Orthogonal Matrix
- Scaling Matrix
- Shear Matrix
- Projection — related but (collapses dimension)