Definition:Quaternion Modulus

Definition

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 $\mathbf x$.

\(\displaystyle \left\vert{\mathbf x}\right\vert\)

\(=\)

\(\displaystyle \sqrt{\det \left({\mathbf x}\right)}\)

\(\displaystyle \)

\(=\)

\(\displaystyle \sqrt{\det \left({\begin{bmatrix} a+bi & c+di \\ -c+di & a-bi \end{bmatrix} }\right)}\)

\(\displaystyle \)

\(=\)

\(\displaystyle \sqrt{\left({a+bi}\right) \left({a-bi}\right) - \left({c+di}\right) \left({-c+di}\right)}\)

Definition of Determinant of Matrix

\(\displaystyle \)

\(=\)

\(\displaystyle \sqrt{\left({a^2 + b^2}\right) - \left({-c^2 - d^2}\right)}\)

\(\displaystyle \)

\(=\)

\(\displaystyle \sqrt{a^2 + b^2 + c^2 + d^2}\)

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 $\mathbf x$.