Definition:Quaternion Modulus

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

Then the (quaternion) modulus of $\mathbf x$ is written as $\vert \mathbf x \vert$ and is defined as:
 * $\vert \mathbf x \vert := \sqrt{a^2 + b^2 + c^2 + d^2}$

The quaternion modulus is a real-valued function, and as when appropriate is referred to as the quaternion modulus function.

Definition 2
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 $\mathbf x$.

Then the (quaternion) modulus of $\mathbf x$ is defined as:
 * $\vert \mathbf x \vert := \sqrt{\det \left({\mathbf x}\right)}$

Equivalence Definition of Quaternion Modulus
Let $\mathbf x = \begin{bmatrix} a+bi & c+di \\ -c+di & a-bi \end{bmatrix}$ be the matrix form of quaternion $\mathbf x$.

Also see

 * Quaternion Modulus is Norm
 * Definition:Modulus of Quaternion-Valued Function
 * Quaternion Modulus in Terms of Conjugate, in which $\left\vert\mathbf x\right\vert$ is defined without having to explicitly reference the components $a,b,c,d$ within $\mathbf x$ as follows:
 * $\left\vert\mathbf x\right\vert:=\sqrt{\mathbf x \overline{\mathbf x} }$
 * where $\overline{\mathbf x}$ denotes the quaternion conjugate of $\mathbf x$.