Norm

The algebra norm of a hypercomplex element: the determinant of multiplication-by-$z$ on the $2^n$-dimensional space. Reduces to the Gaussian $a^2+b^2$ at one imaginary unit.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaNorm (mathematics)↗MathWorldNorm↗WikidataQ956437↗Encyclopedia of Mathematicsnorm↗Britannicatopic/norm-mathematics↗Wolfram LanguageNorm!↗

Norm(z)the determinant of the multiplication-by-$z$ map

Typed signatures

Type	Package	Applies
(list<tuple> | number | tensor | tuple, (+oo | real | string)?) -> +oo | nan | real	compute-engine	always
(list<tuple> | number | tensor | tuple, (+oo | real | string)?) -> +oo | nan | real	hypercomplexreplaces compute-engine	always

Domain: Hypercomplex algebra

Details

• Computed through the tower $A = B[x]/(x^2-\varepsilon)$: for $z = u + xv$, $N(z) = N_B(u^2 - \varepsilon v^2)$, bottoming out at a scalar
• NOT $z\,\overline{z}$ — the signature is mixed, and at $\mathbb{C}_2$ that product keeps a live $i_1i_2$ part
• Multiplicative on a FIXED unit set; the norm is taken over the subalgebra generated by the units occurring in $z$, and adjoining a generator squares it
• $z$ is invertible exactly when its norm is; a norm of $0$ marks a zero divisor
• Left symbolic for the anticommuting families, whose norm is a different object

Examples

Basic #

$3^2+4^2$ — the Gaussian norm

live ↗

In$\left\Vert 3+4i_1\right\Vert$Out$25$

the $4\times 4$ multiplication determinant

live ↗

In$\left\Vert 1+2i_1+3i_1i_2\right\Vert$Out$160$

Properties #

split units give $a^2-b^2$, which is indefinite

live ↗

In$\left\Vert 3+4j_1\right\Vert$Out$-7$

a zero divisor

live ↗

In$\left\Vert 1+i_1i_2\right\Vert$Out$0$

Possible issues #

each factor has norm 2 over $\mathbb{C}_1$, but 4 over $\mathbb{C}_2$

live ↗

In$\left\Vert (1+i_1)(1+i_2)\right\Vert$Out$16$

About

Examples	5
Checked against	Wolfram Language 5 differ
Crosswalk	WikipediaMathWorldWikidataEncyclopedia of MathematicsBritannicaWolfram Language

See also: Conjugate, NonCommutativeMultiply, Basis