Notation
Symbol Meaning \(\mathbf{A}, \mathbf{B}, \mathbf{C}\) matrices and vectors are in bold; \(\mathbf{C} = \mathbf{A}\mathbf{B}\) \(a_{ij}, b_{ij}, c_{ij}\) matrix entries; scalars are in ordinary italics \(\mathbf{I}, \mathbf{J}\) the identity matrix and the rotation matrix that represents \(j\) \(j\) the imaginary unit, \(j^2 = -1\) \(x, y\) and \(u, v\) real and imaginary parts of the two complex numbers \(m_k\) the \(k\)-th product in a multiplication trick
Multiplying two \(2 \times 2\) matrices takes 7 multiplications, not 8. Multiplying two complex numbers takes 3 real multiplications, not 4.
Complex multiplication is a special case of matrix multiplication: a complex number is a special \(2 \times 2\) matrix.
1. A Complex Number as a \(2 \times 2\) Matrix
A complex number is written
\[z = x + jy\]where \(j\) is the imaginary unit, \(j^2 = -1\). The same number is represented by the \(2 \times 2\) matrix
\[z = x + jy \;\longleftrightarrow\; \begin{bmatrix} x & -y \\ y & x \end{bmatrix}\]Splitting the matrix into its diagonal and off-diagonal parts:
\[\begin{bmatrix} x & -y \\ y & x \end{bmatrix} = x \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + y \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = x \mathbf{I} + y \mathbf{J}\]The matrix \(\mathbf{J}\) plays the role of the imaginary unit:
\[\mathbf{J}^2 = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} = -\mathbf{I}\]Therefore, matrix multiplication of these matrices is exactly complex multiplication.
2. 7 Multiplications
For two matrices
\[\mathbf{A} = \begin{bmatrix} a_{00} & a_{01} \\ a_{10} & a_{11} \end{bmatrix}, \qquad \mathbf{B} = \begin{bmatrix} b_{00} & b_{01} \\ b_{10} & b_{11} \end{bmatrix}\]the product \(\mathbf{C} = \mathbf{A}\mathbf{B}\) needs only 7 multiplications:
\[m_1 = a_{00} (b_{01} - b_{11})\] \[m_2 = (a_{00} + a_{01}) b_{11}\] \[m_3 = (a_{10} + a_{11}) b_{00}\] \[m_4 = a_{11} (b_{10} - b_{00})\] \[m_5 = (a_{00} + a_{11})(b_{00} + b_{11})\] \[m_6 = (a_{01} - a_{11})(b_{10} + b_{11})\] \[m_7 = (a_{00} - a_{10})(b_{00} + b_{01})\]The entries of the result are
\[\begin{cases} c_{00} = m_5 + m_4 - m_2 + m_6 \\[4pt] c_{01} = m_1 + m_2 \\[4pt] c_{10} = m_3 + m_4 \\[4pt] c_{11} = m_5 + m_1 - m_3 - m_7 \end{cases}\]3. 3 Multiplications
This is a special case of a \(2 \times 2\) matrix. Three products are enough:
\[m_1 = xu\] \[m_2 = yv\] \[m_3 = (x + y)(u + v)\]The result is
\[(x + yj)(u + vj) = (m_1 - m_2) + (m_3 - m_1 - m_2)\, j\]because the cross term is
\[xv + yu = (x + y)(u + v) - xu - yv = m_3 - m_1 - m_2\]