Definition:Conjugate Quaternion

Definition

Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.

The conjugate quaternion of $\mathbf x$ is defined as:

$\overline {\mathbf x} = a \mathbf 1 - b \mathbf i - c \mathbf j - d \mathbf k$.

Matrix Form

If $\mathbf x$ is defined in matrix form:

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

then:

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

It follows that if:

$\mathbf x = \begin{bmatrix} p & q \\ r & s \end{bmatrix}$

then:

$\overline {\mathbf x} = \begin{bmatrix} s & -q \\ -r & p \end{bmatrix}$

Ordered Pair of Complex Numbers

If $\mathbf x$ is defined as an ordered pair $\left({a, b}\right)$ of complex numbers, then:

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

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): 2.2 The Cayley-Dickson Construction