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

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