Definition:Field Norm of Quaternion
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Definition
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.
Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$.
The norm of $\mathbf x$ is the real number defined as:
- $n \left({\mathbf x}\right) := \left\vert{\mathbf x \overline {\mathbf x} }\right\vert = \left\vert{\overline {\mathbf x} \mathbf x }\right\vert = a^2 + b^2 + c^2 + d^2$
Also known as
Many sources refer to this concept as the norm of $\mathbf x$.
However, it is important to note that the field norm of $\mathbf x$ is not actually a norm as is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ for a general ring or vector space, as it does not satisfy the triangle inequality.
It also needs to be pointed out that not even field norm is a good name, because the quaternions $\mathbb H$ do not even form a field.
This confusing piece of anomalous nomenclature just has to be lived with.
Also see
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem