Definition:Quaternion Modulus/Definition 2

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

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

Also see

 * Equivalence of Definitions of Quaternion Modulus