Definition:Quaternion Modulus
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Definition
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion, where $a, b, c, d \in \R$.
Definition 1
The (quaternion) modulus of $\mathbf x$ is the real-valued function defined and denoted as:
- $\size {\mathbf x} := \sqrt {a^2 + b^2 + c^2 + d^2}$
Definition 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:
- $\size {\mathbf x} := \sqrt {\map \det {\mathbf x} }$
Also see
- Quaternion Modulus in Terms of Conjugate: $\size {\mathbf x} := \sqrt {\mathbf x \overline {\mathbf x} }$ where $\overline {\mathbf x}$ denotes the quaternion conjugate of $\mathbf x$
- Results about the quaternion modulus can be found here.