Matrix Form of Quaternion

Theorem
Let $\mathbf x$ be a quaternion such that:
 * $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$

When the quaternion basis is expressed in the form of matrices:


 * $\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

\qquad \mathbf i = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \qquad \mathbf j = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf k = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$

the general quaternion $\mathbf x$ has the form:
 * $\mathbf x = \begin{bmatrix} a + bi & c + di \\ -c + di & a - bi \end{bmatrix}$

or:
 * $\mathbf x = \begin{bmatrix} w & z \\ -\overline z & \overline w \end{bmatrix}$

where :
 * $w$ and $z$ are complex numbers
 * $\overline z$ is the complex conjugate of $z$.