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