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:

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

That is, if:

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

then:

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