Reflection Matrix

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 $ℝ^{𝑛}$ ).

75.1. Reflection across a line through the origin (2D)

Line making angle $𝜃$ with the $𝑥$ -axis:

𝑅_{𝜃} = [\begin{matrix} cos 2 𝜃 & sin 2 𝜃 \\ sin 2 𝜃 & - cos 2 𝜃 \end{matrix}]

Special cases:

Axis	Matrix
$𝑥$ -axis ( $𝜃 = 0$ )	$[\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$
$𝑦$ -axis ( $𝜃 = \frac{𝜋}{2}$ )	$[\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}]$
Line $𝑦 = 𝑥$ ( $𝜃 = \frac{𝜋}{4}$ )	$[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$
Line $𝑦 = - 𝑥$ ( $𝜃 = - \frac{𝜋}{4}$ )	$[\begin{matrix} 0 & - 1 \\ - 1 & 0 \end{matrix}]$

75.2. Reflection across a hyperplane (general)

For a hyperplane through the origin with unit normal $\hat{𝑛}$ , the reflection is the Householder matrix:

𝑅 = 𝐼 - 2 \hat{𝑛} {\hat{𝑛}}^{𝑇}

This subtracts twice the projection onto $\hat{𝑛}$ from each input.

Example

Reflect across the plane $𝑧 = 0$ in $ℝ^{3}$ . Normal: $\hat{𝑛} = (0, 0, 1)$ .

𝑅 = 𝐼 - 2 [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] (0, 0, 1) = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] - 2 [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 1 \end{matrix}]

$𝑧$ flips sign, $𝑥$ and $𝑦$ preserved. ✓

75.3. Properties

Orthogonal: $𝑅^{𝑇} 𝑅 = 𝐼$ , so $𝑅^{- 1} = 𝑅^{𝑇} = 𝑅$ (reflections are self-inverse — reflecting twice is the identity)
Determinant: $det (𝑅) = - 1$ — orientation-reversing
Eigenvalues: $- 1$ (for the direction perpendicular to the mirror) and $+ 1$ with multiplicity $𝑛 - 1$ (for directions parallel to the mirror)
Preserves lengths and angles (it’s orthogonal)
Involution: $𝑅^{2} = 𝐼$

75.4. Rotations vs reflections

	Rotation	Reflection
$det$	$+ 1$	$- 1$
Orientation	preserved	reversed
Self-inverse?	no (unless $0$ or $𝜋$ )	yes
Symmetric?	no	yes

Composition: two reflections produce a rotation (by $2 𝜃$ , where $𝜃$ is the angle between the mirrors).

75.5. Householder transformations

Reflections of the form $𝑅 = 𝐼 - 2 \vec{𝑣} \frac{{\vec{𝑣}}^{𝑇}}{{\vec{𝑣}}^{𝑇} \vec{𝑣}}$ (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 $det = 0$ (collapses dimension)