# 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

Let $\mathbf x$ be a quaternion defined in matrix form as:

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

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

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

### Ordered Pair of Complex Numbers

Let $\mathbf x$ be a quaternion defined as an ordered pair $\left({a, b}\right)$ of complex numbers.

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

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

## Quaternion Conjugation

The operation of quaternion conjugation is the mapping:

$\overline \cdot: \mathbb H \to \mathbb H: \mathbf x \mapsto \overline{\mathbf x}$.

where $\overline{\mathbf x}$ is the quaternion conjugate of $x$.

That is, it maps a quaternion to its quaternion conjugate.