Complex Numbers

Arithmetic

Addition of complex numbers is defined coordinate-wise:

(a+b⁢i)+(c+d⁢i)=(a+c)+(b+d)⁢i𝑎𝑏𝑖𝑐𝑑𝑖𝑎𝑐𝑏𝑑𝑖(a+bi)+(c+di)=(a+c)+(b+d)i

Multiplication of complex numbers is defined as:

(a+b⁢i)⁢(c+d⁢i)=(a⁢c−b⁢d)+(a⁢d+b⁢c)⁢i𝑎𝑏𝑖𝑐𝑑𝑖𝑎𝑐𝑏𝑑𝑎𝑑𝑏𝑐𝑖(a+bi)(c+di)=(ac-bd)+(ad+bc)i

As Matrices

A complex number a + b ⁢ i 𝑎 𝑏 𝑖 a+bi can be encoded as a 2×2 Matrix like so:

FATAL

This representation has some rather nice properties: multiplication of matrices corresponds to complex multiplication:

(a−bba)⁢(c−ddc)=(a⁢c−b⁢d−(a⁢d+b⁢c)a⁢d+b⁢cc⁢a−b⁢d)matrix𝑎𝑏𝑏𝑎matrix𝑐𝑑𝑑𝑐matrix𝑎𝑐𝑏𝑑𝑎𝑑𝑏𝑐𝑎𝑑𝑏𝑐𝑐𝑎𝑏𝑑\begin{pmatrix}a&-b\\ b&a\\ \end{pmatrix}\begin{pmatrix}c&-d\\ d&c\\ \end{pmatrix}=\begin{pmatrix}ac-bd&-(ad+bc)\\ ad+bc&ca-bd\end{pmatrix}\par

Moreover, Matrix Transposition gives Complex Conjugation:

(a−bba)T=(ab−ba)superscriptmatrix𝑎𝑏𝑏𝑎𝑇matrix𝑎𝑏𝑏𝑎\begin{pmatrix}a&-b\\ b&a\\ \end{pmatrix}^{T}=\begin{pmatrix}a&b\\ -b&a\\ \end{pmatrix}\par