Field Norm of Quaternion is Positive Definite
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Theorem
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 field norm of $\mathbf x$:
- $\map n {\mathbf x} := \cmod {\mathbf x \overline {\mathbf x} }$
Proof
\(\ds \map n {\mathbf x}\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \cmod {\mathbf x \overline {\mathbf x} }\) | \(=\) | \(\ds 0\) | Definition of Field Norm of Quaternion | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds a^2 + b^2 + c^2 + d^2\) | \(=\) | \(\ds 0\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds a = 0, b = 0, c = 0, d = 0\) | \(\) | \(\ds \) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k\) | \(=\) | \(\ds \mathbf 0\) |
Hence $n$ is positive definite by definition.
$\blacksquare$
Sources
- 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $3$. FIELD
- 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