Quaternions

Definition

The existence of complex number presented a question for mathematicians, if a complex number exists in a 2D complex plane, could there be a 3D equivalent?

Sir William Rowan Hamilton among many other mathematicians of the 18th and 19th century had been searching for the answer, Hamilton conjectured that a 3D complex number could be represented by the triple $a + bi + cj$ where $i$ and $j$ are imaginary quantities and square to $-1$, when he was developing the algebra for this triplet the product of them raised a problem when expanded

$$ \begin{align*} z_1 &= a_1 + b_1i + c_1j \\ z_2 &= a_2 + b_2i + c_2j \\ z_1z_2 &= (a_1 + b_1i + c_1j)(a_2 + b_2i + c_2j) \\ &= (a_1b_1 - b_1b_2 - c_1c_2) + (a_1b_2 + b_1a_2)i + (a_1c_2 + c_1a_2)j \\ & \quad + b_1c_2ij + c_1b_2ji \end{align*} $$

The quantities $ij$ and $ji$ represented a problem for Hamilton, even if $ij = -ji$ we are still left with $(b_1c_2 - c_1b_2)ij$

On October 16th, 1843, while he was walking with his wife along the Royal Canal in Ireland he saw the solution as a quadruple instead of a triple, instead of using two imaginary terms, three imaginary terms provided the necessary quantities to resolve products like $ij$

Hamilton defined a quaternion $q$ as

$$ \begin{align*} q = s + ai + bj + ck \quad s,a,b,c \in \mathbb{R} \\ i^2 = j^2 = k^2 = ijk = -1 \\ ij = k \quad jk = i \quad ki = j \\ ji = -k \quad kj = -1 \quad ik = -j \end{align*} $$

If a complex number $i$ is capable of rotating points on the plane by $\deg{90}$ then perhaps a triple rotates points in space by $\deg{90}$, in the end the triplet was replaced by a quaternion

Notation

There are three ways of annotating a quaternion $q$

$$ \begin{align} q &= s + xi + yj + zk \\ q &= s + \mbold{v} \\ q &= [s, \mbold{v}] \\ & \text{where $s,x,y,z \in \mathbb{R}$, $\mbold{v} \in \mathbb{R}^3$} \nonumber \\ & \text{and $i^2 = j^2 = k^2 = ijk = -1$} \nonumber \end{align} $$

Real quaternion

A real quaternion has a zero vector term

$$ q = [s, \mbold{0}] $$

Pure quaternion

A pure quaternion is a quaternion having a zero scalar term

$$ q = [0, \mbold{v}] $$

Quaternion conjugate

Given

$$ q = [s, \mbold{v}] $$

The quaternion conjugate is defined as

$$ q^* = [s, - \mbold{v}] $$

Quaternion norm

The norm of a quaternion $q = [s, \mbold{v}]$ is defined as the square root of the product of itself and its conjugate (the multiplication operation is defined later)

$$ \begin{align*} \norm{q} &= \sqrt{qq^*} \\ &= \sqrt{s^2 + x^2 + y^2 + z^2} \end{align*} $$

Also note that

$$ \norm{q}^2 = qq^* $$

Norm facts

$\norm{qq^*} = \norm{q}\norm{q^*}$
$\norm{q^*} = \norm{q}$

Unit quaternion

A unit quaternion is a quaternion of norm one given by

$$ \begin{align} q &= [s, \lambda \unit{n}] \quad s,\lambda \in \mathbb{R}, \unit{n} \in \mathbb{R}^3 \label{unit-norm-quaternion}\\ \left | \unit{n} \right | &= 1 \nonumber \\ s^2 + \lambda^2 &= 1 \nonumber \end{align} $$

Note: dividing a non-zero quaternion by its norm produces a unit norm quaternion

Operations

Quaternion Product

Given two quaternions

$$ \begin{align*} q_a = [s_a, \mbold{a}] \quad \quad \mbold{a} = x_a i + y_a j + z_a k \\ q_b = [s_b, \mbold{b}] \quad \quad \mbold{b} = x_b i + y_b j + z_b k \end{align*} $$

The product $q_aq_b$ is computed as follows

$$ \begin{align} q_aq_b &= (s_a + x_a i + y_a j + z_a k)(s_b + x_b i + y_b j + z_b k) \nonumber \\ &= (s_as_b - x_ax_b - y_ay_b - z_az_b) \nonumber \\ & \quad + (s_ax_b + s_bx_a + y_az_b - y_bz_a)i \nonumber \\ & \quad + (s_ay_b + s_by_a + z_ax_b - z_bx_a)j \nonumber \\ & \quad + (s_az_b + s_bz_a + x_ay_b - x_by_a)k \label{quaternion-product} \end{align} $$

Replacing the imaginaries by the ordered pairs (which are themselves quaternion units)

$$ i = [0, \mbold{i}] \quad j = [0, \mbold{j}] \quad k = [0, \mbold{k}] \quad 1 = [1, \mbold{0}] $$

And substituting them in \eqref{quaternion-product}

$$ \begin{align*} q_aq_b &= (s_as_b - x_ax_b - y_ay_b - z_az_b)[1, \mbold{0}] \\ & \quad + (s_ax_b + s_bx_a + y_az_b - y_bz_a)[0, \mbold{i}] \\ & \quad + (s_ay_b + s_by_a + z_ax_b - z_bx_a)[0, \mbold{j}] \\ & \quad + (s_az_b + s_bz_a + x_ay_b - x_by_a)[0, \mbold{k}] \end{align*} $$

By doing some aggrupations

$$ \begin{align*} q_aq_b &= [s_as_b - x_ax_b - y_ay_b - z_az_b, \\ & \quad s_a(x_b \mbold{i} + y_b \mbold{j} + z_b \mbold{k}) + s_b(x_a \mbold{i} + y_a \mbold{j} + z_a \mbold{k}) \\ & \quad + (y_az_b - y_bz_a) \mbold{i} + (z_ax_b - z_bx_a) \mbold{j} + (x_ay_b - x_by_a) \mbold{k}] \\ &= [s_as_b - \mbold{a} \cdot \mbold{b}, s_a\mbold{b} + s_b\mbold{a} + \mbold{a} \times \mbold{b}] \end{align*} $$

Now let’s compute the product $q_bq_a$

$$ q_bq_a = [s_bs_a - \mbold{b} \cdot \mbold{a}, s_b\mbold{a} + s_a\mbold{b} + \mbold{b} \times \mbold{a}] $$

Note that the scalar quantity of both products is the same however the vector quantity varies (the cross product sign is changed) therefore

$$ q_aq_b \not = q_bq_a $$

This is an important fact to note since for complex number the product commutes however for quaternions it doesn’t

Product of a scalar and a quaternion

Let $k$ be a scalar represented as a quaternion as $q_k = [k, \mathbf{0}]$ and $q = [s, \mathbf{v}]$

Their product is

$$ \begin{align*} q_kq &= [k, \mathbf{0}][s, \mathbf{v}] \\ &= [ks, k\mathbf{v}] \end{align*} $$

Note that this product is commutative

Product of a quaternion with itself (square of a quaternion)

$$ \begin{align*} q &= [s, \mbold{v}] \\ q^2 &= [s, \mbold{v}] [s, \mbold{v}] \\ &= [s^2 - \mbold{v} \cdot \mbold{v}, 2s\mbold{v} + \mbold{v} \times \mbold{v}] \\ &= [s^2 - \norm{v}^2, 2s\mbold{v}] \\ &= [s^2 - (x^2 + y^2 + z^2), 2s(x\mbold{i} + y\mbold{j} + z\mbold{k})] \end{align*} $$

Product of a quaternion and its conjugate

Let $q = [s, \mathbf{v}]$

$$ \begin{align*} qq^* &= [s, \mathbf{v}][s, -\mathbf{v}] \\ &= [s^2 + \mathbf{v} \cdot \mathbf{v}, -s \mathbf{v} + s\mathbf{v} - \mathbf{v} \times \mathbf{v}] \\ &= [s^2 + \mathbf{v} \cdot \mathbf{v}, \mathbf{0}] \\ &= s^2 + x^2 + y^2 + z^2 \end{align*} $$

Note that this product commutes i.e. $qq^* = q^*q$

Product of unit quaternions

Given

$$ q_a = [s_a, \mbold{a}] \\ q_b = [s_b, \mbold{b}] $$

Where $\norm{q_a} = \norm{q_b} = 1$, the product is another unit-norm quaternion

$$ q_c = [s_c, \mbold{c}] $$

Where $\norm{q_c} = 1$

Product of pure quaternions

Let

$$ q_a = [0, \mbold{a}] \\ q_b = [0, \mbold{b}] $$

The product $q_aq_b$ is defined as

$$ \begin{align*} q_aq_b &= [-\mbold{a} \cdot \mbold{b}, \mbold{a} \times \mbold{b}] \end{align*} $$

Note that the resulting quaternion is no longer a pure quaternion as some information has propagated into the real part via the dot product

Product of a pure quaternion with itself (square of a pure quaternion)

$$ \begin{align*} q &= [0, \mbold{v}] \\ q^2 &= [0, \mbold{v}] [0, \mbold{v}] \\ &= [-\mbold{v} \cdot \mbold{v}, \mbold{v} \times \mbold{v}] \\ &= [-(x^2 + y^2 + z^2), \mbold{0}] \\ &= -\norm{v}^2 \end{align*} $$

If $q$ is a unit norm pure quaternion then

$$ q^2 = -1 $$

Product of a pure quaternion with its conjugate

$$ \begin{align*} q^*q = qq^* &= [0, \mathbf{v}][0, -\mathbf{v}] \\ &= [\mathbf{v} \cdot \mathbf{v}, -\mbold{v \times v}] \\ &= [\mathbf{v} \cdot \mathbf{v}, \mbold{0}] \\ &= \norm{v}^2 \end{align*} $$

Inverse of a quaternion

By definition, the inverse $q^{-1}$ of $q$ is

$$ qq^{-1} = [1, \mbold{0}] $$

To isolate $q^{-1}$ let’s pre multiply both sides by $q^*$

$$ \begin{align*} q^*qq^{-1} &= q^* \\ \norm{q}^2q^{-1} &= q^* \\ q^{-1} &= \frac{q^*}{\norm{q}^2} \end{align*} $$

Quaternion units

Given the vector $\mbold{v}$

$$ \mbold{v} = v \hat{\mbold{v}} \quad \text{where $v = |\mbold{v}|$, and $|\hat{\mbold{v}}| = 1$} $$

Combining this with the definition of a pure quaternion

$$ \begin{align*} q &= [0, \mbold{v}] \\ &= [0, v \hat{\mbold{v}}] \\ &= v[0, \hat{\mbold{v}}] \end{align*} $$

It’s convenient to identify the unit quaternion as $\hat{q}$ (where $v = 1$)

$$ \hat{q} = [0, \hat{\mbold{v}}] $$

Let’s check if the quaternion unit $\mbold{i}$ squares to the ordered pair $[-1, \mbold{0}]$

$$ \begin{align*} i^2 &= [0, \mbold{i}][0, \mbold{i}] \\ &= [0 \cdot 0 - \mbold{i} \cdot \mbold{i}, 0 \cdot \mbold{i} + 0 \cdot \mbold{i} - \mbold{i} \times \mbold{i}] \\ &= [-|\mbold{i}|^2, \mbold{0}] \quad \text{$\mbold{i} \times \mbold{i} = 0$} \\ & = [-1, \mbold{0}] \end{align*} $$

Misc operations

Taking the scalar part of a quaternion

To isolate the scalar part of $q$ we could add $q^*$ to it

$$ 2 S(q) = q + q^* $$