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Quaternion

Multiplication of two quaterions

$$ab = (a_s b_s - a_v \cdot b_v) + (a_s b_v + a_v b_s + a_v \times b_v)$$

\begin{align} ab = &(a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4) \

• &(a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)i \
• &(a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2)j \
• &(a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1)k \end{align}

Matrix Representaion

Define

$$A(a) = \begin{bmatrix} a_1 & -a_2 & -a_3 & -a_4 \\ a_2 & a_1 & -a_4 & a_3 \\ a_3 & a_4 & a_1 & -a_2 \\ a_4 & -a_3 & a_2 & a_1 \end{bmatrix}$$

where $a$, $b$, $c$, and $d$ are the components of a quaterion.

$$A = \begin{bmatrix} a_1 & -a_2 & -a_3 & -a_4 \\ a_2 & a_1 & -a_4 & a_3 \\ a_3 & a_4 & a_1 & -a_2 \\ a_4 & -a_3 & a_2 & a_1 \end{bmatrix} \begin{bmatrix} b_1 \\ b_2 \\ b_3 \\ b_4 \end{bmatrix}$$

Multiplication

If we have quaternion multiplication

$$ab = c$$

this can be written as matrix-vector multiplication

$$A b = c$$