Quaternion Modulus of Conjugate
Jump to navigation
Jump to search
Theorem
Let $z = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.
Let $\overline z$ be the conjugate of $z$.
Let $\cmod z$ be the quaternion modulus of $z$.
Then:
- $\cmod {\overline z} = \cmod z$
Proof
\(\ds \cmod z\) | \(=\) | \(\ds a^2 + b^2 + c^2 + d^2\) | Definition of Quaternion Modulus | |||||||||||
\(\ds \cmod {\overline z}\) | \(=\) | \(\ds \cmod {a \mathbf 1 - b \mathbf i - c \mathbf j - d \mathbf k}\) | Definition of Quaternion Conjugate | |||||||||||
\(\ds \) | \(=\) | \(\ds a^2 + \paren {-b}^2 + \paren {-c}^2 + \paren {-d}^2\) | Definition of Quaternion Modulus | |||||||||||
\(\ds \) | \(=\) | \(\ds a^2 + b^2 + c^2 + d^2\) |
$\blacksquare$