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

Proof
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