[Math] Show that a reflection matrix is given by $\begin{bmatrix}\cos2\theta&\sin2\theta \\ \sin2\theta&-\cos2\theta\end{bmatrix}$

linear algebramatricesproof-verificationproof-writing

Reflection matrix:

$$ \text{Reflection}(\theta) =
\begin{bmatrix}
\cos2\theta & \sin2\theta \\
\sin2\theta & -\cos2\theta
\end{bmatrix}$$

Attempt:

Inspiration:
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Speaking non-rigorously, it seems like the angle between the reflected vector and the original vector will be $2\theta$. Armed with this, let's consider how $e_1 = \begin{bmatrix}1\\0\end{bmatrix}$ and $e_2 = \begin{bmatrix}0\\1\end{bmatrix}$ change when we reflect them across an arbitrary line.

Let $$
\text{Reflection}(\theta) =
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}$$

Then,

$$\begin{align}
\text{Reflection}(\theta) \cdot e_1 &=
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix} \cdot \begin{bmatrix}1\\0\end{bmatrix} \\
&= \begin{bmatrix}a\\c\end{bmatrix}
\end{align}$$

Using my assumption that reflected vectors have angle of $2\theta$ between itself and the original vector, $(\text{Reflection}(\theta)\cdot e_1) \cdot e_1 = \begin{bmatrix}a\\c\end{bmatrix}\cdot \begin{bmatrix}1\\0\end{bmatrix} = a = ||\text{Reflection}(\theta)\cdot e_1||\cdot||e_1|| \cos(2\theta)$ (dot product).

Simplifying the other side of the equation, we get:
$$a = 1\cdot 1 \cos(2\theta) = \cos(2\theta)$$

Doing it similarly for $e_2$ yields $d=\cos(2\theta)$ which means our reflection matrix currently looks like this:

$$ \text{Reflection}(\theta) =
\begin{bmatrix}
\cos2\theta & b \\
c & \cos2\theta
\end{bmatrix}$$

which is not correct.

Questions:

  1. I suspect my implicit assumption that a reflection can be represented as a rotation of $2\theta$, where $\theta$ is the angle between the original vector and reflection line, is where I went wrong. Why is this wrong?

$$ \text{Rotation}(2\theta) =
\begin{bmatrix}
\cos2\theta & -\sin2\theta \\
\sin2\theta & \cos2\theta
\end{bmatrix} \text{ for reference. }$$

  1. What's the correct way to do this?

I would normally ask one question per SE question but I think my two questions are tightly coupled.

Best Answer

The problem with your attempted derivation is that you keep changing the definition of $\theta$ since "the original vector" can be any input to the linear transformation.

Presumably, the $\theta$ in the definition of the reflection is such that the axis of reflection is along the vector $(\cos \theta, \sin \theta)$, which is to say that $\theta$ is the angle between the axis of rotation and the vector $e_1$. With that in mind, your derivation of the entry $a$ is fine, but your derivation of the entry $d$ is unjustified and ultimately incorrect.

Here's a geometric approach involving rotations that you might like. Let $F_\theta$ denote the matrix of the reflection through the axis at angle $\theta$, and let $R_\theta$ denote the matrix of the counterclockwise rotation by $\theta$. Note that the matrix of the reflection through the axis at $\theta = 0$ is given by $$ F_0 = \pmatrix{1&0\\0&-1}. $$ On the other hand, we can implement a reflection through the axis at angle $\theta$ by first rotating that axis to $\theta = 0$, then applying $F_0$, then rotating the axis back. In other words, we have $$ F_\theta = R_{\theta} F_0 R_{-\theta} = \pmatrix{\cos \theta & - \sin \theta\\ \sin \theta & \cos \theta} \pmatrix{1&0\\0&-1} \pmatrix{\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta}. $$ After multiplying and simplifying with the double-angle laws, you should indeed find that $$ F_\theta = \pmatrix{\cos 2 \theta & \sin 2 \theta \\ \sin 2 \theta & - \cos 2 \theta}. $$

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