1.14: Representing Complex Multiplication as Matrix Multiplication

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Consider two complex number \(z_1 = a + bi\) and \(z_2 = c + di\) and their product

\[z_1 z_2 = (a + bi) (c + id) = (ac - bd) + i(bc + ad) =: \omega \label{eq3} \]

Now let's define two matrices

\[Z_1 = \begin{bmatrix}a & -b \\ b & a \end{bmatrix} \nonumber \]

\[Z_2 = \begin{bmatrix} c & -d \\ d & c \end{bmatrix} \nonumber \]

Note that these matrices store the same information as \(z_1\) and \(z_2\), respectively. Let’s compute their matrix product

\[Z_1 Z_2 = \begin{bmatrix} a & -b \\ b & a \end{bmatrix} \begin{bmatrix} c & -d \\ d & c \end{bmatrix} = \begin{bmatrix} ac - bd & -(bc + ad) \\ bc + ad & ac - bd \end{bmatrix} := W. \nonumber \]

Comparing \(W\) just above with \(w\) in Equation \ref{eq3}, we see that \(W\) is indeed the matrix corresponding to the complex number \(w = z_1 z_2\). Thus, we can represent any complex number \(z\) equivalently by the matrix

\[Z = \begin{bmatrix} \text{Re} z & -\text{Im} z \\ \text{Im} z & \text{Re} z \end{bmatrix} \nonumber \]

and complex multiplication then simply becomes matrix multiplication. Further note that we can write

\[Z = \text{Re} z \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \text{Im} z \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, \nonumber \]

i.e., the imaginary unit \(i\) corresponds to the matrix \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\) and \(i^2 = -1\) becomes

\[\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = -\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}. \nonumber \]

Polar Form (Decomposition)

Writing \(z = re^{i \theta} = r(\cos \theta + i \sin \theta)\), we find

\[ \begin{align*} Z &= r \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \\[4pt] &= \begin{bmatrix} r & 0 \\ 0 & r \end{bmatrix} \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \end{align*} \]

corresponding to a stretch factor \(r\) multiplied by a 2D rotation matrix. In particular, multiplication by \(i\) corresponds to the rotation with angle \(\theta = \pi /2\) and \(r = 1\).

We will not make a lot of use of the matrix representation of complex numbers, but later it will help us remember certain formulas and facts.