Nothing new under the sun

So, I had a random idea about representing complex numbers as commuting matrices, which comes out in sort of a neat way. I spent an hour or so typing up some simple results of this representation...and then, on a hunch that I was (as usual) reinventing the wheel, I gingerly typed "complex numbers as matrices" into Google.

Yup, this representation is so well-known that it's on Wikipedia. Sigh...oh well. But, since I already typed this up, I thought I'd tell the world all about one more way in which Pericles reinvents the wheel!

A complex number $z = x+iy$ is written as an ordered pair of real numbers $x$ and $y$, which corresponds to a vector in the complex plane:
\[ z = (x,y) = \begin{bmatrix} \, x \, \\ \, y \, \end{bmatrix}. \]
Addition of complex numbers is carried out as ordinary vector addition,
\[ z_1 + z_2 = \begin{bmatrix} \, x_1 \, \\ \, y_1 \, \end{bmatrix} + \begin{bmatrix} \, x_2 \, \\ \, y_2 \, \end{bmatrix} = \begin{bmatrix} \, x_1 + x_2 \, \\ \, y_1 + y_2 \, \end{bmatrix}. \]
Multiplication of complex numbers uses $i^2 = -1$ and is carried out as
\[ z_1 z_2 = (x_1+iy_1)(x_2+iy_2) = (x_1x_2-y_1y_2)+i(x_1y_2+y_1x_2) = \begin{bmatrix} \, x_1x_2-y_1y_2 \, \\ \, y_1x_2+y_2x_1 \, \end{bmatrix}. \]
This can be rewritten as a matrix product,
\[ \begin{bmatrix} \, x_1x_2-y_1y_2 \, \\ \, y_1x_2+y_2x_1 \, \end{bmatrix} = \begin{bmatrix} \, x_1 \, \\ \, y_1 \, \end{bmatrix}x_2 + \begin{bmatrix} \, -y_1 \, \\ \, x_1 \, \end{bmatrix} y_2 = \begin{bmatrix} \, x_1 & -y_1 \, \\ \, y_1 & x_1 \, \end{bmatrix} \begin{bmatrix} \, x_2 \, \\ \, y_2 \, \end{bmatrix}. \]
These matrices commute, as required:
\[ z_1 z_2 = \begin{bmatrix} \, x_1 & -y_1 \, \\ \, y_1 & x_1 \, \end{bmatrix} \begin{bmatrix} \, x_2 \, \\ \, y_2 \, \end{bmatrix} = \begin{bmatrix} \, x_2 & -y_2 \, \\ \, y_2 & x_2 \, \end{bmatrix} \begin{bmatrix} \, x_1 \, \\ \, y_1 \, \end{bmatrix} = z_2 z_1. \]
Because a complex number has more structure than an ordinary vector, a more complete picture is given by thinking of a complex number as a matrix. That is,
\[ z = x + iy \implies \begin{bmatrix} \, x & -y \, \\ \, y & x \, \end{bmatrix} \equiv Z, \]
where $Z$ denotes the matrix form of the complex number $z$. Using this notation, addition is
\[ Z_1 + Z_2 = \begin{bmatrix} \, x_1 & -y_1 \, \\ \, y_1 & x_1 \, \end{bmatrix} + \begin{bmatrix} \, x_2 & -y_2 \, \\ \, y_2 & x_2 \, \end{bmatrix} = \begin{bmatrix} \, x_1 + x_2 & -(y_1+y_2) \, \\ \, y_1+y_2 & x_1+x_2 \, \end{bmatrix}, \]
and multiplication is
\[ Z_1 Z_2 = \begin{bmatrix} \, x_1 & -y_1 \, \\ \, y_1 & x_1 \, \end{bmatrix} \begin{bmatrix} \, x_2 & -y_2 \, \\ \, y_2 & x_2 \, \end{bmatrix} = \begin{bmatrix} \, x_1x_2-y_1y_2 & -(y_1x_2+y_2x_1) \, \\ \, y_1x_2+y_2x_1 & x_1x_2-y_1y_2 \, \end{bmatrix}. \]
The inverse of $z$,
\[z^{-1} = \left( \frac{x}{x^2 + y^2}, \frac{-y}{x^2+y^2} \right), \]
is found simply from the inverse of the matrix $Z$:
\[Z^{-1} = \frac{1}{x^2+y^2} \begin{bmatrix} \, x & y \, \\ \, -y & x \, \end{bmatrix}.\]

The invariants of the matrix $Z$ are related in simple ways to quantities of interest associated with the complex number $z$. The rank of $Z$, $\operatorname{rank}(Z) = 2$, indicates that the complex plane is two-dimensional. The modulus of $z$ is the square root of the determinant of $Z$:
\[ |z| = \sqrt{\det (Z)} = \sqrt{x^2 + y^2}. \]
The real part of $z = x+iy$, $\Re(z) = \Re(x+iy) = x$, is calculated from the trace of $Z$,
\[ \Re(z) = \frac{1}{2} \operatorname{tr} \left(Z\right) = x. \]
Similar to the sum form of $z = x+iy$, the matrix $Z$ may be written as a sum of a symmetric and skew-symmetric matrix,
\[ Z = x \begin{bmatrix} \, 1 & 0 \, \\ \, 0 & 1 \, \end{bmatrix} + y \begin{bmatrix} \, 0 & -1 \, \\ \, 1 & 0 \, \end{bmatrix} = x + yC, \]
where $C$ is the complementary operator, the matrix form of the imaginary number $i$, and $x$ multiplies the identity matrix, the matrix form of $1$.

Evidently, these matrices $\{ 1, C \}$ are the generating elements of the matrix representation of the field of complex numbers. That is, they play a role equivalent to that of basis vectors in an ordinary vector space: every matrix representation of a complex number can be written as a linear combination of these 'basis matrices.' This is interesting, because the set of $2\times 2$ matrices forms a ring, but not a field; matrix multiplication does not in general commute.