Matrix Representations
$$ \newcommand{\N}{n} \newcommand{\K}{k} \newcommand{\H}[1]{h_{#1}} \newcommand{\cosk}[1]{\operatorname{\mathfrak{c}_{#1}}} \newcommand{\sink}[1]{\operatorname{\mathfrak{s}_{#1}}} \newcommand{\tank}[1]{\operatorname{\mathfrak{t}_{#1}}} $$This page documents a few interesting number systems along with their matrix representations.
Preliminaries
The unified treatments below make use of generalized trigonometric functions.
Complex Number Family
Initially, we consider a few select members of the complex number family. In the process, we highlight parellels between them. Subsequently, we provide a unified perspective.
Individual Members
Lets consider how to represent the complex numbers, and two of their close relatives.
Complex Numbers
Cartesian Forms
Matrices $\mathbf{Z} \in \mathbb{R}^{2\times 2}$ of the form: $$\mathbf{Z} = \mathbf{1}a + \mathbf{I}b = \left[\begin{matrix}a & b\\-b & a\end{matrix}\right]$$ where: $$ \begin{align*} \mathbf{1} &= \left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right] \\\\ \mathbf{I} &= \left[\begin{matrix}0 & 1\\-1 & 0\end{matrix}\right] \end{align*} $$ then: $$ \mathbf{I}^2 = \left[\begin{matrix}-1 & 0\\0 & -1\end{matrix}\right] = -\mathbf{1} $$
Operations
Given $\mathbf{Z} = \mathbf{1}a + \mathbf{I}b$ we have:
Addition
$$\small \mathbf{Z}_0 + \mathbf{Z}_1 = \left[\begin{matrix}a_0+a_1 & b_0+b_1\\-(b_0+b_1) & a_0+a_1\end{matrix}\right]$$Additive Inverse
$$-\mathbf{Z} = \left[\begin{matrix}-a & -b\\+b & -a\end{matrix}\right]$$Multiplication
$$\small \mathbf{Z}_0 \mathbf{Z}_1 = \left[\begin{matrix}a_0 a_1 - b_0 b_1 & a_0 b_1 + a_1 b_0\\-(a_0 b_1 + a_1 b_0) & a_0 a_1 - b_0 b_1\end{matrix}\right] $$Conjugate
$$\overline{\mathbf{Z}} = adj(\mathbf{Z}) = \left[\begin{matrix}a & -b\\+b & a\end{matrix}\right]$$Quadratic Form
$$|\mathbf{Z}|^2 = det(\mathbf{Z}) = a^2 + b^2$$Multiplicative Inverse
$$\mathbf{Z}^{-1} = \frac{1}{|\mathbf{Z}|^2} \overline{\mathbf{Z}}$$Polar/Exponential Forms
Substituting $\mathbf{I}\theta$ for $x$ in $e^{x}$: $$ \begin{align} e^{\mathbf{I}\theta} &= \mathbf{1}\cos(\theta) + \mathbf{I}\sin(\theta) \\\\ &= \left[\begin{matrix}\cos(\theta) & \sin(\theta)\\-\sin(\theta) & \cos(\theta)\end{matrix}\right] \end{align} $$ which motivates: $$ \begin{align} re^{\mathbf{I}\theta} &= r(\mathbf{1}\cos(\theta) + \mathbf{I}\sin(\theta))\\\\ &= \left[\begin{matrix}r\cos(\theta) & r\sin(\theta)\\ -r\sin(\theta) & r\cos(\theta)\end{matrix}\right] \end{align} $$
Dual Numbers
Cartesian Forms
Matrices $\mathbf{Z} \in \mathbb{R}^{2\times 2}$ of the form: $$\mathbf{Z} = \mathbf{1}a + \mathbf{E}b = \left[\begin{matrix}a & b\\0 & a\end{matrix}\right]$$ where: $$ \begin{align*} \mathbf{1} &= \left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right] \\\\ \mathbf{E} &= \left[\begin{matrix}0 & 1\\0 & 0\end{matrix}\right] \end{align*} $$ then: $$ \mathbf{E}^2 = \left[\begin{matrix}0 & 0\\0 & 0\end{matrix}\right] = \mathbf{0} $$
Operations
Given $\mathbf{Z} = \mathbf{1}a + \mathbf{E}b$ we have:
Addition
$$\small \mathbf{Z}_0 + \mathbf{Z}_1 = \left[\begin{matrix}a_0+a_1 & b_0+b_1\\0 & a_0+a_1\end{matrix}\right]$$Additive Inverse
$$-\mathbf{Z} = \left[\begin{matrix}-a & -b\\0 & -a\end{matrix}\right]$$Multiplication
$$\small \mathbf{Z}_0 \mathbf{Z}_1 = \left[\begin{matrix}a_0 a_1 & a_0 b_1 + a_1 b_0\\0 & a_0 a_1\end{matrix}\right] $$Conjugate
$$\overline{\mathbf{Z}} = adj(\mathbf{Z}) = \left[\begin{matrix}a & -b\\0 & a\end{matrix}\right]$$Quadratic Form
$$|\mathbf{Z}|^2 = det(\mathbf{Z}) = a^2$$Multiplicative Inverse
$$\mathbf{Z}^{-1} = \frac{1}{|\mathbf{Z}|^2} \overline{\mathbf{Z}}$$Polar/Exponential Forms
Substituting $\mathbf{E}\theta$ for $x$ in $e^{x}$: $$ \begin{align} e^{\mathbf{E}\theta} &= \mathbf{1} + \mathbf{E}\theta \\\\ &= \left[\begin{matrix}1 & \theta\\0 & 1\end{matrix}\right] \end{align} $$ which motivates: $$ \begin{align} re^{\mathbf{E}\theta} &= r(\mathbf{1} + \mathbf{E}\theta)\\\\ &= \left[\begin{matrix}r & r\theta\\0 & r\end{matrix}\right] \end{align} $$
Split-Complex Numbers
Cartesian Forms
Matrices $\mathbf{Z} \in \mathbb{R}^{2\times 2}$ of the form: $$\mathbf{Z} = \mathbf{1}a + \mathbf{J}b = \left[\begin{matrix}a & b\\+b & a\end{matrix}\right]$$ where: $$ \begin{align*} \mathbf{1} &= \left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right] \\\\ \mathbf{J} &= \left[\begin{matrix}0 & 1\\+1 & 0\end{matrix}\right] \end{align*} $$ then: $$ \mathbf{J}^2 = \left[\begin{matrix}+1 & 0\\0 & +1\end{matrix}\right] = +\mathbf{1} $$
Operations
Given $\mathbf{Z} = \mathbf{1}a + \mathbf{J}b$ we have:
Addition
$$\small \mathbf{Z}_0 + \mathbf{Z}_1 = \left[\begin{matrix}a_0+a_1 & b_0+b_1\\+(b_0+b_1) & a_0+a_1\end{matrix}\right]$$Additive Inverse
$$-\mathbf{Z} = \left[\begin{matrix}-a & -b\\-b & -a\end{matrix}\right]$$Multiplication
$$\small \mathbf{Z}_0 \mathbf{Z}_1 = \left[\begin{matrix}a_0 a_1 + b_0 b_1 & a_0 b_1 + a_1 b_0\\+(a_0 b_1 + a_1 b_0) & a_0 a_1 + b_0 b_1\end{matrix}\right] $$Conjugate
$$\overline{\mathbf{Z}} = adj(\mathbf{Z}) = \left[\begin{matrix}a & -b\\-b & a\end{matrix}\right]$$Quadratic Form
$$|\mathbf{Z}|^2 = det(\mathbf{Z}) = a^2 - b^2$$Multiplicative Inverse
$$\mathbf{Z}^{-1} = \frac{1}{|\mathbf{Z}|^2} \overline{\mathbf{Z}}$$Polar/Exponential Forms
Substituting $\mathbf{J}\theta$ for $x$ in $e^{x}$: $$ \begin{align} e^{\mathbf{J}\theta} &= \mathbf{1}\cosh(\theta) + \mathbf{J}\sinh(\theta) \\\\ &= \left[\begin{matrix}\cosh(\theta) & \sinh(\theta)\\ \sinh(\theta) & \cosh(\theta)\end{matrix}\right] \end{align} $$ which motivates: $$ \begin{align} re^{\mathbf{J}\theta} &= r(\mathbf{1}\cosh(\theta) + \mathbf{J}\sinh(\theta))\\\\ &= \left[\begin{matrix}r\cosh(\theta) & r\sinh(\theta)\\ r\sinh(\theta) & r\cosh(\theta)\end{matrix}\right] \end{align} $$
Unified Family
The similarities in the matrix representations above suggest a possible unification as follows.
Unified Complex Numbers
Cartesian Form
Matrices $\mathbf{Z} \in \mathbb{R}^{2\times 2}$ of the form: $$\mathbf{Z} = \mathbf{1}a + \mathbf{K}b = \left[\begin{matrix}a & b\\-kb & a\end{matrix}\right]$$ where: $$ \begin{align*} \mathbf{1} &= \left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right] \\\\ \mathbf{K} &= \left[\begin{matrix}0 & 1\\-k & 0\end{matrix}\right] \end{align*} $$ then: $$ \mathbf{K}^2 = \left[\begin{matrix}-k & 0\\0 & -k\end{matrix}\right] = -\mathbf{1}k $$
Operations
Given $\mathbf{Z} = \mathbf{1}a + \mathbf{K}b$ we have:
Addition
$$\small \mathbf{Z}_0 + \mathbf{Z}_1 = \left[\begin{matrix}a_0+a_1 & b_0+b_1\\-\K(b_0+b_1) & a_0+a_1\end{matrix}\right]$$Additive Inverse
$$-\mathbf{Z} = \left[\begin{matrix}-a & -b\\ \K b & -a\end{matrix}\right]$$Multiplication
$$\small \mathbf{Z}_0 \mathbf{Z}_1 = \left[\begin{matrix}a_0 a_1 - \K b_0 b_1 & a_0 b_1 + a_1 b_0\\-\K(a_0 b_1 + a_1 b_0) & a_0 a_1 - \K b_0 b_1\end{matrix}\right] $$Conjugate
$$\overline{\mathbf{Z}} = adj(\mathbf{Z}) = \left[\begin{matrix}a & -b\\ \K b & a\end{matrix}\right]$$Quadratic Form
$$|\mathbf{Z}|^2 = det(\mathbf{Z}) = a^2 + \K b^2$$Multiplicative Inverse
$$\mathbf{Z}^{-1} = \frac{1}{|\mathbf{Z}|^2} \overline{\mathbf{Z}}$$Polar/Exponential Form
Substituting $\mathbf{K}\theta$ for $x$ in $e^{x}$: $$ \begin{align} e^{\mathbf{K}\theta} &= \mathbf{1}\mathfrak{c}_{k}(\theta) + \mathbf{K}\mathfrak{s}_{k}(\theta) \\\\ &= \left[\begin{matrix}\mathfrak{c}_{k}(\theta) & \mathfrak{s}_{k}(\theta)\\-k\mathfrak{s}_{k}(\theta) & \mathfrak{c}_{k}(\theta)\end{matrix}\right] \end{align} $$ which motivates: $$ \begin{align} re^{\mathbf{K}\theta} &= r(\mathbf{1}\mathfrak{c}_{k}(\theta) + \mathbf{K}\mathfrak{s}_{k}(\theta))\\\\ &= \left[\begin{matrix}r\mathfrak{c}_{k}(\theta) & r\mathfrak{s}_{k}(\theta)\\ -kr\mathfrak{s}_{k}(\theta) & r\mathfrak{c}_{k}(\theta)\end{matrix}\right] \end{align} $$
In other words, the above exhibits an algebra homomorphism from the algebra of general complex numbers $\mathbb{C}_{k}$ to the algebra of real matrices $\mathbb{R}^{2\times 2}$.
Quaternion Family
Lets review the quaternions, and two of their close relatives.
Individual Members
Now lets see how to represent these numbers using matrices.
Unified Family
Now lets see how to represent these numbers using matrices.
Unified Quaternions
Cartesian Form
Matrices $\mathbf{Z} \in \mathbb{R}^{4\times 4}$ of the form: $$ \begin{align} \mathbf{Z} &= \mathbf{1}a + \mathbf{I}b + \mathbf{J}c + \mathbf{K}d \\\\ &= \left[\begin{matrix} a & b & c & d \\ -\H0 b & a & -\H0 d & c \\ -\H1 c & \H1 d & a & -b \\ -\H0 \H1 d & -\H1 c & \H0 b & a \end{matrix}\right] \end{align} $$ where: $$ \begin{align*} \mathbf{1} &= \left[\begin{matrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{matrix}\right] \\\\ \mathbf{I} &= \left[\begin{matrix}0 & +1 & 0 & 0\\-h_0 & 0 & 0 & 0\\0 & 0 & 0 & -1\\0 & 0 & +h_0 & 0\end{matrix}\right] \\\\ \mathbf{J} &= \left[\begin{matrix}0 & 0 & +1 & 0\\0 & 0 & 0 & +1\\-h_1 & 0 & 0 & 0\\0 & -h_1 & 0 & 0\end{matrix}\right] \\\\ \mathbf{K} &= \left[\begin{matrix}0 & 0 & 0 & +1\\0 & 0 & -h_0 & 0\\0 & +h_1 & 0 & 0\\ -h_0 h_1 & 0 & 0 & 0\end{matrix}\right] \end{align*} $$ then: $$ \begin{align*} \mathbf{I}^2 &= -\mathbf{1}h_0 \\\\ \mathbf{J}^2 &= -\mathbf{1}h_1 \\\\ \mathbf{K}^2 &= -\mathbf{1}h_0 h_1 \end{align*} $$
Properties
Given $\mathbf{Z} = \mathbf{1}a + \mathbf{I}b + \mathbf{J}c + \mathbf{K}d$ we have:
Addition
$$\small \mathbf{Z}_0 + \mathbf{Z}_1 = \left[\begin{matrix} a_0+a_1 & \cdots \\ -\H0 (b_0+b_1) & \cdots \\ -\H1 (c_0+c_1) & \cdots \\ -\H0 \H1 (d_0+d_1) & \cdots \end{matrix}\right]$$Additive Inverse
$$-\mathbf{Z} = \left[\begin{matrix} -a & -b & -c & -d \\ \H0 b & -a & \H0 d & -c \\ \H1 c & -\H1 d & -a & b \\ \H0 \H1 d & \H1 c & -\H0 b & -a \end{matrix}\right]$$Multiplication
$$\small \mathbf{Z}_0 \mathbf{Z}_1 = \left[\begin{matrix} a_0 a_1 - \H0 b_0 b_1 - \H1 c_0 c_1 - \H0 \H1 d_0 d_1 & \cdots \\ -\H0 (a_0 b_1 + a_1 b_0 + \H1 c_0 d_1 - \H1 c_1 d_0) & \cdots \\ -\H1 (a_0 c_1 + a_1 c_0 - \H0 b_0 d_1 + \H0 b_1 d_0) & \cdots \\ -\H0 \H1 (a_0 d_1 + a_1 d_0 + b_0 c_1 - b_1 c_0) & \cdots \end{matrix}\right] $$Conjugate
$$ \overline{\mathbf{Z}} = \frac{adj(\mathbf{Z})}{|\mathbf{Z}|^2} = \left[\begin{matrix} a & -b & -c & -d \\ \H0 b & a & \H0 d & -c \\ \H1 c & -\H1 d & a & b \\ \H0 \H1 d & \H1 c & -\H0 b & a \end{matrix}\right] $$Quadratic Form
$$|\mathbf{Z}|^2 = \frac{det(\mathbf{Z})}{|\mathbf{Z}|^2} = a^2 + h_0 b^2 + h_1 c^2 + h_0 h_1 d^2$$Multiplicative Inverse
$$\mathbf{Z}^{-1} = \frac{1}{|\mathbf{Z}|^2} \overline{\mathbf{Z}}$$Polar/Exponential Form
Substituting $\mathbf{\hat{N}}\theta$ for $x$ in $e^{x}$: $$ \begin{align} e^{\mathbf{\hat{N}}\theta} &= \mathbf{1}\cosk{\N}(\theta) + \mathbf{\hat{N}}\sink{\N}(\theta) \\\\ &= \left[\begin{matrix} \cosk{\N}(\theta) & b \sink{\N}(\theta) & c \sink{\N}(\theta) & d \sink{\N}(\theta) \\ -\H0 b \sink{\N}(\theta) & \cosk{\N}(\theta) & -\H0 d \sink{\N}(\theta) & c \sink{\N}(\theta) \\ -\H1 c \sink{\N}(\theta) & +\H1 d \sink{\N}(\theta) & \cosk{\N}(\theta) & -b \sink{\N}(\theta) \\ -\H0 \H1 d \sink{\N}(\theta) & -\H1 c \sink{\N}(\theta) & +\H0 b \sink{\N}(\theta) & \cosk{\N}(\theta) \end{matrix}\right] \end{align} $$ where $n = |\mathbf{\hat{N}}|^2$ and $\mathbf{\hat{N}}$ is pure imaginary: $$ \begin{align} \mathbf{\hat{N}} &= \mathbf{I}b + \mathbf{J}c + \mathbf{K}d \\\\ &= \left[\begin{matrix} 0 & b & c & d \\ -\H0 b & 0 & -\H0 d & c \\ -\H1 c & +\H1 d & 0 & -b \\ -\H0 \H1 d & -\H1 c & +\H0 b & 0 \end{matrix}\right] \end{align} $$ which motivates: $$ \begin{align} re^{\mathbf{\hat{N}}\theta} &= r(\mathbf{1}\cosk{\N}(\theta) + \mathbf{\hat{N}}\sink{\N}(\theta))\\\\ &= r\left[\begin{matrix} \cosk{\N}(\theta) & b \sink{\N}(\theta) & c \sink{\N}(\theta) & d \sink{\N}(\theta) \\ -\H0 b \sink{\N}(\theta) & \cosk{\N}(\theta) & -\H0 d \sink{\N}(\theta) & c \sink{\N}(\theta) \\ -\H1 c \sink{\N}(\theta) & +\H1 d \sink{\N}(\theta) & \cosk{\N}(\theta) & -b \sink{\N}(\theta) \\ -\H0 \H1 d \sink{\N}(\theta) & -\H1 c \sink{\N}(\theta) & +\H0 b \sink{\N}(\theta) & \cosk{\N}(\theta) \end{matrix}\right] \end{align} $$
In other words, the above exhibits an algebra homomorphism from the algebra of general quaternions $\mathbb{H}_{\H0,\H1}$ to the algebra of real matrices $\mathbb{R}^{4\times 4}$.