Quaternion Modulus of Product of Quaternions

Theorem

Let $\mathbf x, \mathbf y$ be quaternions.

Let $\size {\mathbf x}$ be the modulus of $\mathbf x$.

Then:

$\size {\mathbf {x y} } = \size {\mathbf x} \size {\mathbf y}$

Proof

Let $\mathbf x, \mathbf y$ be in their matrix form.

Then:

\(\ds \size {\mathbf {x y} }\)

\(=\)

\(\ds \sqrt {\map \det {\mathbf {x y} } }\)

Definition of Quaternion Modulus

\(\ds \)

\(=\)

\(\ds \sqrt {\map \det {\mathbf x} \map \det {\mathbf y} }\)

Determinant of Matrix Product

\(\ds \)

\(=\)

\(\ds \sqrt {\map \det {\mathbf x} } \sqrt {\map \det {\mathbf y} }\)

Exponent Combination Laws

\(\ds \)

\(=\)

\(\ds \size {\mathbf x} \size {\mathbf y}\)

Definition of Quaternion Modulus

$\blacksquare$