Complex Matrix Expansions

I was proud of this idea, as it was probably the best early research idea I've come up with.

I became very interested in matrices and linear algebra after reading a paper on modelling origami using rotation and translation matrices (Belcastro and Hull, 2012). I began to play with expressing all kinds of things as matrices and seeing what "meanings" matrix operations had in those contexts.

Somehow complex numbers cropped up, and I decided that they were a good candidate for this "interpretation" because multiplying by a complex number meant a rotation and dilation of the complex plane - or an "amplitwist" (Needham, 1996). So I represented a complex number as a rotation matrix together with a scaling factor.

Given a complex number $z = r\mathrm{e}^{\mathrm{i}\theta}$, let $\mathbf{Z} = r\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}.$ Some interesting properties follow immediately from this correspondence:

1. If $z = 0$ then $\mathbf{Z} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}.$ (zero matrix)
   
2. If $z = 1$ then $\mathbf{Z} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}.$ (identity matrix)
3. $z^{-1}$ corresponds to $\mathbf{Z}^{-1}.$ (matrix inverse)
4. $z^\ast$ (complex conjugate) corresponds to $\mathbf{Z}^\intercal.$ (matrix transpose)
5. $\det(\mathbf{Z}) = \lvert z\rvert^2.$ (matrix determinant, absolute value).

I liked to write (5) as  $\lvert\mathbf{Z}\rvert = \lvert z\rvert^2$, using the "absolute value" notation for matrix determinants.

In terms of Abstract Algebra, the next two properties are more important. Suppose we have complex numbers $a$ and $b$ that correspond to matrices $\mathbf{A}$ and $\mathbf{B}$ respectively.

• $ab$ corresponds to $\mathbf{A}\mathbf{B}.$ (homomorphic on multiplication)
• $a + b$ corresponds to $\mathbf{A} + \mathbf{B}.$ (homomorphic on addition)

The multiplication property comes directly from how I defined the "matrix correspondence": by treating matrix multiplication as a matrix transformation. The addition property was unexpected, but on hindsight it came from the fact that a complex number $x + y\mathrm{i}$ would correspond to $\left[\begin{smallmatrix}x & -y \\ y & x\end{smallmatrix}\right].$

Prove all of the properties mentioned above. How would you generalize this "expansion" of a complex number further? I have some hints if you're stuck...

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