Quaternion Modulus of Conjugate

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

\(\ds \)

\(=\)

\(\ds a^2 + b^2 + c^2 + d^2\)

$\blacksquare$