Function: Octonion[onorm] - norm of an octonion

Calling Sequence:

onorm(o);

Parameters:

o - expression of the type 'octonion'

Description:

Procedure 'onorm' calculates norm of an octonion o. It is defined as follows:

onorm(o) = sqrt(o &o o_conjug(o)) = sqrt(x0^2+x1^2+ x2^2+x3^2+x4^2+x5^2+x6^2+x7^2)

where o = x0+x1*e1+x2*e2+x3*e3+x4*e4+x5*e5+x6*e6+x7*e7, and x0,x1,...,x7, are real parameters.

Recall that octonionic product can be computed with the procedure Octonion[omul] or with its infix form `&o`.

For information about type 'octonion' see Octonion[`type/octonion`] .

Examples:

> restart:with(Cliff5):with(Octonion);

Warning, new definition for init

> o1:=1-2*e1+3*e3+e4-e6+e7;

> onorm(o1); #norm of o1

> o2:=2-3*e4+e5+4*e6-e7;

> onorm(o2);

Theoreom [The Eight-Square Identity]

The norm in the octonion algebra is a ring homomorphism.

> o1:=x0+x1*e1+x2*e2+x3*e3+x4*e4+x5*e5+x6*e6+x7*e7;

> o2:=y0+y1*e1+y2*e2+y3*e3+y4*e4+y5*e5+y6*e6+y7*e7;

We will now verify that

onorm(o1 &o o2) = onorm(o1) * onorm(o2).

> onorm(o1 &o o2);

> factor(%);

> onorm(o1)*onorm(o2);

>

See Also: Octonion[oversion] , Octonion[omul] , Octonion[oinv] , Octonion[def_omultable] , Octonion[omultable]