Motivation
The complex numbers allow us to model rotations in the plane as multiplication of the elements in it. A more fundamental operation in the plane is that of reflection, since we can write rotations and translations as compositions of it but not vice versa. I wish to construct a set of number system starting from a geometric standpoint which will allow me to describe reflection as multiplication between elements in it. The inspiration for this attempt was mostly given by viewing videos of Geometric Algebra on the internet. Especially Steven De Kenick's talk on Dual Quarternions.
Viewing Complex numbers as matrices
Consider the imaginary unit $i$, the action of it on the position vector $\begin{bmatrix} x \\ y \end{bmatrix}$ when represented as a complex number is given as:
$$ i(x+iy) = ix-y$$
The above equation, we could think of it in matrix form with vectors as:
$$i \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -y \\ x \end{bmatrix} $$
One could guess that the matrix form of $i$ takes the form:
$$ i= \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$$
Hence, $i$'s action on a complex number is equivalent to $M$'s action on a vector.
The standpoint we will take hence forth will be to derive the Reflection Numbers (putting faith they do exist) by beginning with a matrix.
Deriving the matrices for reflection number
We can write down the following properties by seeing the geometry effect of reflections:
- $$J^2=I$$
- $$J=J^{-1}$$
- $$ \text{det} J = -1$$
Explanation: 1. Doing the reflection twice is same as not doing anything , 2. By multiplying both side of 1. by $J^{-1}$ , 3. Due to the fact that reflection reverses orientation of area.
Let's assume $$J= \begin{bmatrix} a&b \\ c & d \end{bmatrix}$$ , Then by adjoint equation, we find it's inverse is given as:
$$J^{-1} = \frac{1}{\text{det} J} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = (-1) \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \begin{bmatrix} -d & b \\ c & -a \end{bmatrix} $$
Since $J^{-1}=J$, we find that $d=-a$. Furthermore $M^{-1} = M^T$ since $M$ preserves dot the product, meaning that $c=b$
We also know that $\text{det} J=-1=-a^2 - b^2$ or:
$$ a^2 + b^2-1=0$$
We can parameterize the above circle as $a= \cos \theta $ , $ b = \sin \theta$. Meaning $$M(\theta) = \begin{bmatrix} \cos \theta & \sin \theta \\ \sin \theta & - \cos \theta \end{bmatrix}$$
Now, consider suppose this keeps a line through origin having direction $\begin{bmatrix} \cos \phi \\ \sin \phi \end{bmatrix}$ as fixed then:
$$ \begin{bmatrix} \cos \theta & \sin \theta \\ \sin \theta & - \cos \theta \end{bmatrix} \begin{bmatrix} \cos \phi \\ \sin \phi \end{bmatrix} = \begin{bmatrix} \cos \phi \\ \sin \phi \end{bmatrix}$$
By multiplication we find:
$$\begin{bmatrix} \cos (\theta- \phi) \\ \sin (\theta-\phi) \end{bmatrix} = \begin{bmatrix} \cos \phi \\ \sin \phi \end{bmatrix}$$
Hence, to keep the direction vector at angle $\phi$ fixed, our matrix need to be evaluated at $\theta=2\phi$. We haven now:
$$M(2 \phi)= \begin{bmatrix} \cos 2 \phi & \sin 2 \phi \\ \sin 2 \phi & - \cos 2\phi\end{bmatrix}$$
At this point I split $M$ into two using these two different matrices(*):
$$ e_1 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, e_2 = \begin{bmatrix} 0&1 \\ 1&0 \end{bmatrix}$$ making
$$M_{2 \phi} = e_1 \cos 2 \phi + e_2 \sin 2 \phi$$
Where $e_1^2 = e_2^2 =1$ and $e_2 e_1 = i =-e_2 e_1$ (where $i$ is in the matrix sense)
Proof that matrix $M$ actually reflects points not line which makes angle $\phi$
Let the point $P$'s position vector be decomposed into component along the reflecting line $P_l e_l$ and the along a line perpendicular to it $P_n e_n$, then:
$$M_{2\phi} P = M_{2\phi} (P_l e_l + P_n e_n) = P_l M_{2\phi} e_l + P_n M_{2\phi} e_n= P_le_l- P_n n$$
Since the reflection matrix reverse the sign of normal. The above is exactly what we expect when we reflect hence QED.
My questions on the approach
*: Is this choice unique? I am really taken aback by the decomposition here actually since $e_1$ is a reflection along x-axis and $e_2$ is reflection along line $y=x$, why must these two be the building block of all other reflections?
**: Is there a name for this number system? (if it were studied as it's own thing before)
***: What deeper things could I do with the results found?