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