Definition:Conjugate Quaternion/Matrix Form

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

It follows that if:
 * $\mathbf x = \begin{bmatrix} p & q \\ r & s \end{bmatrix}$

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