Definition:Quaternion

Definition

A quaternion is a number in the form:

$a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$

where:

• $a, b, c, d$ are real numbers;

• $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ are entities related to each other in the following way:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \mathbf i \mathbf j = - \mathbf j \mathbf i\)	\(=\)	\(\displaystyle \mathbf k\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \mathbf j \mathbf k = - \mathbf k \mathbf j\)	\(=\)	\(\displaystyle \mathbf i\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \mathbf k \mathbf i = - \mathbf i \mathbf k\)	\(=\)	\(\displaystyle \mathbf j\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \mathbf i^2 = \mathbf j^2 = \mathbf k^2 = \mathbf i \mathbf j \mathbf k\)	\(=\)	\(\displaystyle - \mathbf 1\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

The set of all quaternions is usually denoted $\mathbb H$.

Quaternion Addition

The sum of two quaternions $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ is defined as:

$\mathbf x_1 + \mathbf x_2 := \left({a_1 + a_2}\right) \mathbf 1 + \left({b_1 + b_2}\right) \mathbf i + \left({c_1 + c_2}\right) \mathbf j + \left({d_1 + d_2}\right) \mathbf k$

Quaternion Multiplication

The product of two quaternions $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ is defined as:

	\(\displaystyle \)	\(\displaystyle \mathbf x_1 \mathbf x_2\)	\(\displaystyle \)	\(\displaystyle   :=\)	\(\)	\(\displaystyle \left({a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2}\right) \mathbf 1\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(+\)	\(\displaystyle \left({a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2}\right) \mathbf i\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(+\)	\(\displaystyle \left({a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2}\right) \mathbf j\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(+\)	\(\displaystyle \left({a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2}\right) \mathbf k\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

This is proved to be consistent with the definition of quaternions in Quaternion Multiplication.

Construction from Complex Matrices

Let $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ denote the following four elements of the matrix space $\mathcal M_\C \left({2}\right)$:

$\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad \mathbf i = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \qquad \mathbf j = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf k = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$

where $\C$ is the set of complex numbers.

The set:

$Q_4 = \left\{{\mathbf 1, -\mathbf 1, \mathbf i, -\mathbf i, \mathbf j, -\mathbf j, \mathbf k, -\mathbf k}\right\}$

under the operation of conventional matrix multiplication, forms the quaternion group:

$\begin{array}{c|cccccccc} & \mathbf 1 & \mathbf i & -\mathbf 1 & -\mathbf i & \mathbf j & \mathbf k & -\mathbf j & -\mathbf k \\ \hline \mathbf 1 & \mathbf 1 & \mathbf i & -\mathbf 1 & -\mathbf i & \mathbf j & \mathbf k & -\mathbf j & -\mathbf k \\ \mathbf i & \mathbf i & -\mathbf 1 & -\mathbf i & \mathbf 1 & \mathbf k & -\mathbf j & -\mathbf k & \mathbf j \\ -\mathbf 1 & -\mathbf 1 & -\mathbf i & \mathbf 1 & \mathbf i & -\mathbf j & -\mathbf k & \mathbf j & \mathbf k \\ -\mathbf i & -\mathbf i & \mathbf 1 & \mathbf i & -\mathbf 1 & -\mathbf k & \mathbf j & \mathbf k & -\mathbf j \\ \mathbf j & \mathbf j & -\mathbf k & -\mathbf j & \mathbf k & -\mathbf 1 & \mathbf i & \mathbf 1 & -\mathbf i \\ \mathbf k & \mathbf k & \mathbf j & -\mathbf k & -\mathbf j & -\mathbf i & -\mathbf 1 & \mathbf i & \mathbf 1 \\ -\mathbf j & -\mathbf j & \mathbf k & \mathbf j & -\mathbf k & \mathbf 1 & -\mathbf i & -\mathbf 1 & \mathbf i \\ -\mathbf k & -\mathbf k & -\mathbf j & \mathbf k & \mathbf j & \mathbf i & \mathbf 1 & -\mathbf i & -\mathbf 1 \end{array}$

In Quaternions Defined by Matrices it is shown that these have the appropriate properties as defined above.

In Matrix Form of Quaternion it is shown that a general element $\mathbf x$ of $\mathbb H$ has the form:

$\mathbf x = \begin{bmatrix} a + bi & c + di \\ -c + di & a - bi \end{bmatrix}$

Construction from Complex Pairs

A quaternion can be defined as an ordered pair $\left({x, y}\right)$ where $x, y \in \C$ are complex numbers, on which the operations of addition and multiplication are defined as follows:

Quaternion Addition of Complex Pairs

Let $x_1, x_2, y_1, y_2$ be complex numbers.

Then $\left({x_1, y_1}\right) + \left({x_2, y_2}\right)$ is defined as:

$\left({x_1, y_1}\right) + \left({x_2, y_2}\right):= \left({x_1 + x_2, y_1 + y_2}\right)$

Quaternion Multiplication of Complex Pairs

Let $x_1, x_2, y_1, y_2$ be complex numbers.

Then $\left({x_1, y_1}\right) \left({x_2, y_2}\right)$ is defined as:

$\left({x_1, y_1}\right) \left({x_2, y_2}\right) := \left({x_1 x_2 - y_2 \overline {y_1}, \overline {x_1} y_2 + y_1 x_2}\right)$

where $\overline {x_1}$ and $\overline {y_1}$ are the complex conjugates of $x_1$ and $y_1$ respectively.

From Quaternions Defined by Ordered Pairs this definition can be seen to be equivalent to the main definition above.

Construction from Cayley-Dickson Construction

The quaternions can be defined by the Cayley-Dickson construction from the set of complex numbers $\C$.

From Complex Numbers form Algebra, $\C$ forms a nicely normed $*$-algebra.

Let $a, b \in \C$.

Then $a, b \in \Bbb H$, where:

$\left({a, b}\right) \left({c, d}\right) = \left({a c - d \overline b, \overline a d + c b}\right)$

$\overline {\left({a, b}\right)} = \left({\overline a, -b}\right)$

where:

$\overline a$ is the complex conjugate of $a$

and

$\overline {\left({a, b}\right)}$ is the conjugation operation on $\Bbb H$.

It is clear by direct comparison with the Construction from Complex Pairs that this construction genuinely does generate the Quaternions.

Quaternion Algebra Over a Field

A quaternion algebra can be defined over any field as follows:

Let $\mathbb K$ be a field, and $a$, $b \in \mathbb K$.

Define the quaternion algebra $\left\langle{ a,b }\right\rangle_\mathbb K$ to be the $\mathbb K$-vector space with basis $\{1, i, j, k\}$ subject to:

• $i^2 = a$
• $j^2 = b$
• $ij = k = -ji$

Formally this could be achieved as a multiplicative presentation of a suitable group, or as a linear subspace of a finite extension of $\mathbb K$.

Taking $\mathbb K = \R$ and $a = b = -1$ we see than this generalises Hamilton's quaternions above.

Also see

In Ring of Quaternions it is shown that $\mathbb H$ forms a ring under the operations of conventional matrix addition and matrix multiplication.

In Quaternions Subring of Complex Matrix Space it is shown that $\mathbb H$ is a subring of the matrix space $\mathcal M_\C \left({2}\right)$.

In Quaternions form Skew Field is it shown that $\mathbb H$ actually forms a skew field under the operations of conventional matrix addition and matrix multiplication.

In Complex Numbers Subfield of Quaternions it is shown that $\C$ is isomorphic to a subfield of $\mathbb H$.

Alternative notation

Some sources use $V$ for $\mathbb H$.

History

The quaternions were famously conceived by William Rowan Hamilton, who was so proud of his flash of insight that he carved:

$i^2 = j^2 = k^2 = i j k = -1$

into the stone of Brougham Bridge on October 16, 1843.

Linguistic Note

The word quaternion is derived from the Latin word quaterni, meaning four by four.

The word quaternion is also used for a style of poem in which the theme is divided into four complementary parts.

It's an awkward word - the fingers keep trying to type it as quaternian which, although it feels more natural, is technically incorrect.

Sources

• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.2$: Ring Example $9$
• John C. Baez: The Octonions (2002): 1 Introduction