Equivalence of Definitions of Quaternion Modulus

Theorem

Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion, where $a, b, c, d \in \R$.

The following definitions of the concept of Quaternion Modulus are equivalent:

The (quaternion) modulus of $\mathbf x$ is the real-valued function defined and denoted as:

$\size {\mathbf x} := \sqrt {a^2 + b^2 + c^2 + d^2}$

Let $\mathbf x$ be expressed in matrix form:

$\mathbf x = \begin {bmatrix} a + b i & c + d i \\ -c + d i & a - b i \end {bmatrix}$

The (quaternion) modulus of $\mathbf x$ is the real-valued function defined and denoted as:

$\size {\mathbf x} := \sqrt {\map \det {\mathbf x} }$

Let $\mathbf x = \begin{bmatrix} a + b i & c + d i \\ -c + d i & a - b i \end{bmatrix}$ be the matrix form of quaternion $\mathbf x$.

$\ds \size {\mathbf x}$

$=$

$\ds \sqrt {\map \det {\mathbf x} }$

$\ds $

$=$

$\ds \sqrt {\map \det {\begin{bmatrix} a + b i & c + d i \\ -c + d i & a - b i \end{bmatrix} } }$

$\ds $

$=$

$\ds \sqrt {\paren {a + b i} \paren {a - b i} - \paren {c + d i} \paren {-c + d i} }$

$\ds $

$=$

$\ds \sqrt {\paren {a^2 + b^2} - \paren {-c^2 - d^2} }$

$\ds $

$=$

$\ds \sqrt {a^2 + b^2 + c^2 + d^2}$

$\blacksquare$