Octonion Conjugation is Involution
Jump to navigation
Jump to search
Theorem
Let $x = \tuple {a, b}: a, b \in \mathbb H$ be a octonion.
Let $\overline x$ be the conjugate of $x$.
Then:
- $\overline \cdot: x \mapsto \overline x$
is an involution.
That is:
- $\overline {\paren {\overline x} } = x$
Proof
\(\ds \overline {\paren {\overline x} }\) | \(=\) | \(\ds \overline {\paren {\overline {\tuple {a, b} } } }\) | Definition of Octonion | |||||||||||
\(\ds \) | \(=\) | \(\ds \overline {\tuple {\overline a, -b} }\) | Definition of Conjugate of Octonion | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {\overline {\paren {\overline a} }, -\paren {-b} }\) | Definition of Conjugate of Octonion | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {a, b}\) | Quaternion Conjugation is Involution | |||||||||||
\(\ds \) | \(=\) | \(\ds x\) | Definition of Octonion |
$\blacksquare$