Reflection
Chapter 6: Geometric Transformation and Eigen Decomposition — Reflection
From the book
Chapter 6: Geometric Transformation and Eigen Decomposition. In the chapter mind map this icon labels Reflection. The discussion below is excerpted and lightly edited from § Reflection in Mathematics for AI and Machine Learning.
In 2D, a reflection matrix reflects vectors across a line through the origin. For reflection across a line making angle $\theta$ with the $x$-axis:
For reflection across the $x$-axis: $F = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$
For reflection across the $y$-axis: $F = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$
As shown in the book figure a 2D reflection transformation flips the object across an axis. Reflection across the $x$-axis (red dashed line) and $y$-axis (green dash-dot line) both preserve the area but reverse the orientation of the object.
- $F^2 = I$ (reflecting twice returns to original)
- $\det(F) = -1$ (reverses orientation)
What this drawing shows
What you see. Black reference square fixed; red dashed copy flips across the x-axis from the identity to the mirrored position (y scale $1 \to -1$), matching Householder reflection geometry.
In the mind map. Chapter 6 — Reflection. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 6: Geometric Transformation and Eigen Decomposition — see the mind-map node Reflection.