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
- 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 $\mathbf x$.