Product of Quaternion Conjugates
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Theorem
Let $\mathbf x, \mathbf y \in \mathbb H$ be quaternions.
Let $\overline{\mathbf x}$ be the conjugate of $\mathbf x$.
Then:
- $\overline{\mathbf x \times \mathbf y} = \overline{\mathbf y} \times \overline{\mathbf x}$
but in general:
- $\overline{\mathbf x \times \mathbf y} \ne \overline{\mathbf x} \times \overline{\mathbf y}$
Proof
Consider the matrix form of $\mathbf x$ and $\mathbf y$:
- $\mathbf x = \begin{bmatrix} a & b \\ -\overline b & \overline a \end{bmatrix}$
- $\mathbf y = \begin{bmatrix} c & d \\ -\overline d & \overline c \end{bmatrix}$
where $a, b, c, d \in \C$.
\(\ds \overline{\mathbf x \times \mathbf y}\) | \(=\) | \(\ds \overline{\begin{bmatrix} a & b \\ -\overline b & \overline a \end{bmatrix} \begin{bmatrix} c & d \\ -\overline d & \overline c \end{bmatrix} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \overline{\begin{bmatrix} a c - b \overline d & a d + b \overline c \\ - \overline b c - \overline a \overline d & - \overline b d + \overline a \overline c \end{bmatrix} }\) | Definition of Matrix Product (Conventional) | |||||||||||
\(\ds \) | \(=\) | \(\ds \begin{bmatrix} \overline{a c - b \overline d} & -\left({a d + b \overline c}\right) \\ \overline b c + \overline a \overline d & \overline{- \overline b d + \overline a \overline c} \end{bmatrix}\) | Definition of Conjugate Quaternion | |||||||||||
\(\ds \) | \(=\) | \(\ds \begin{bmatrix} \overline a \overline c - \overline b d & - a d - b \overline c \\ \overline {a d} + \overline b c & a c - b \overline d \end{bmatrix}\) | Complex Conjugation is Automorphism, Complex Conjugation is Involution |
\(\ds \overline{\mathbf y} \times \overline{\mathbf x}\) | \(=\) | \(\ds \overline{\begin{bmatrix} c & d \\ -\overline d & \overline c \end{bmatrix} } \ \overline{\begin{bmatrix} a & b \\ -\overline b & \overline a \end{bmatrix} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \begin{bmatrix} \overline c & - d \\ \overline d & c \end{bmatrix} \begin{bmatrix} \overline a & - b \\ \overline b & a \end{bmatrix}\) | Definition of Conjugate Quaternion | |||||||||||
\(\ds \) | \(=\) | \(\ds \begin{bmatrix} \overline a \overline c - \overline b d & - a d - b \overline c \\ \overline {a d} + \overline b c & a c - b \overline d \end{bmatrix}\) | Definition of Matrix Product (Conventional) | |||||||||||
\(\ds \) | \(=\) | \(\ds \overline{\mathbf x \times \mathbf y}\) | from above |
but:
\(\ds \overline{\mathbf x} \times \overline{\mathbf y}\) | \(=\) | \(\ds \overline{\begin{bmatrix} a & b \\ -\overline b & \overline a \end{bmatrix} } \ \overline{\begin{bmatrix} c & d \\ -\overline d & \overline c \end{bmatrix} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \begin{bmatrix} \overline a & - b \\ \overline b & a \end{bmatrix} \times \begin{bmatrix} \overline c & - d \\ \overline d & c \end{bmatrix}\) | Definition of Conjugate Quaternion | |||||||||||
\(\ds \) | \(=\) | \(\ds \begin{bmatrix} \overline a \overline c - b \overline d & - \overline a d - b c \\ a \overline d + \overline {b c} & a c - \overline b d \end{bmatrix}\) | Definition of Matrix Product (Conventional) | |||||||||||
\(\ds \) | \(\ne\) | \(\ds \overline{\mathbf x \times \mathbf y}\) | from above |
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem