Definition:Quaternion Modulus

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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

$\left\vert\mathbf x\right\vert:=\sqrt{\mathbf x \overline{\mathbf x} }$
where $\overline{\mathbf x}$ denotes the quaternion conjugate of $\mathbf x$.