Field Norm of Quaternion is Positive Definite

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 $\mathbf x$:

$\map n {\mathbf x} := \cmod {\mathbf x \overline {\mathbf x} }$

$\ds \map n {\mathbf x}$

$=$

$\ds 0$

$\ds \leadstoandfrom \ \ $

$\ds \cmod {\mathbf x \overline {\mathbf x} }$

$=$

$\ds 0$

$\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.

$\blacksquare$

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