Quaternion Multiplication
Jump to navigation
Jump to search
Theorem
Let $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ be quaternions.
Then their product is given by:
| \(\ds \mathbf x_1 \mathbf x_2 \ \ \) | \(\ds =\) | \(\) | \(\ds \left({a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2}\right) \mathbf 1\) | |||||||||||
| \(\ds \) | \(+\) | \(\ds \left({a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2}\right) \mathbf i\) | ||||||||||||
| \(\ds \) | \(+\) | \(\ds \left({a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2}\right) \mathbf j\) | ||||||||||||
| \(\ds \) | \(+\) | \(\ds \left({a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2}\right) \mathbf k\) |
Proof
From Matrix Form of Quaternion we have that:
- $\mathbf x_1 \mathbf x_2 = \begin{bmatrix} a_1 + b_1 i & c_1 + d_1 i \\ -c_1 + d_1 i & a_1 - b_1 i \end{bmatrix} \begin{bmatrix} a_2 + b_2 i & c_2 + d_2 i \\ -c_2 + d_2 i & a_2 - b_2 i \end{bmatrix}$
Let $\mathbf x_1 \mathbf x_2 = \begin{bmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{bmatrix} = \begin{bmatrix} a + b i & c + d i \\ -c + d i & a - b i \end{bmatrix}$.
Throughout we use the definition of conventional matrix product.
So:
| \(\ds p_{11}\) | \(=\) | \(\ds \left({a_1 + b_1 i}\right) \left({a_2 + b_2 i}\right) + \left({c_1 + d_1 i}\right) \left({-c_2 + d_2 i}\right)\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds a_1 a_2 + b_1 a_2 i + a_1 b_2 i + b_1 b_2 i^2 - c_1 c_2 - d_1 c_2 i + c_1 d_2 i + d_1 d_2 i^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \left({a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2}\right) + \left({a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2}\right) i\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds a + b i\) |
| \(\ds p_{12}\) | \(=\) | \(\ds \left({a_1 + b_1 i}\right) \left({c_2 + d_2 i}\right) + \left({c_1 + d_1 i}\right) \left({a_2 - b_2 i}\right)\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds a_1 c_2 + b_1 c_2 i + a_1 d_2 i + b_1 d_2 i^2 + c_1 a_2 + d_1 a_2 i - c_1 b_2 i - d_1 b_2 i^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \left({a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2}\right) + \left({a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2}\right) i\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds c + d i\) |
| \(\ds p_{21}\) | \(=\) | \(\ds \left({-c_1 + d_1 i}\right) \left({a_2 + b_2 i}\right) + \left({a_1 - b_1 i}\right) \left({-c_2 + d_2 i}\right)\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -c_1 a_2 + d_1 a_2 i - c_1 b_2 i + d_1 b_2 i^2 - a_1 c_2 + b_1 c_2 i + a_1 d_2 i - b_1 d_2 i^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\left({a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2}\right) + \left({a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2}\right) i\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -c + d i\) |
| \(\ds p_{22}\) | \(=\) | \(\ds \left({-c_1 + d_1 i}\right) \left({c_2 + d_2 i}\right) + \left({a_1 - b_1 i}\right) \left({a_2 - b_2 i}\right)\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -c_1 c_2 + d_1 c_2 i - c_1 d_2 i + d_1 d_2 i^2 + a_1 a_2 - b_1 a_2 i - a_1 b_2 i + b_1 b_2 i^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \left({a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2}\right) - \left({a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2}\right) i\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds a - b i\) |
Hence the result from Matrix Form of Quaternion.
$\blacksquare$