find the norm of a quaternion or octonion - Maple Help

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LieAlgebras[AlgebraNorm] - find the norm of a quaternion or octonion

LieAlgebras[AlgebraInverse] - find the multiplicative inverse of a quaternion or octonion 

Calling Sequences

    AlgebraNorm(X)

    AlgebraInverse(X)

Parameters

     X   - a quaternion or octonion

     

 

Description

Examples

Description

• 

If X  is a quaternion or octonion then the norm of is X = XX  where X  is the conjugate of X.

• 

The inverse of X is X1 = X _/X2.

• 

For example, if X is the quaternion X=a + bi + cj +dk, then X  = a  bi  cj dk, X = a2 +b2  +c2  +d2 and X1 =  a bi  cj dka2 +b2  +c2  +d2.

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

Use AlgebraLibraryData to retrieve the structure equations for the quaternions. Call the algebra Qalg.

ADAlgebraLibraryDataQuaternions,Qalg

AD:=e12=e1,e1.e2=e2,e1.e3=e3,e1.e4=e4,e2.e1=e2,e22=e1,e2.e3=e4,e2.e4=e3,e3.e1=e3,e3.e2=e4,e32=e1,e3.e4=e2,e4.e1=e4,e4.e2=e3,e4.e3=e2,e42=e1

(2.1)

 

Initialize the algebra of quaternions.

DGsetupAD,'e','i','j','k','α'

algebra name: Qalg

(2.2)

 

Define a quaternion X.

Qalg > 

X2e+3i4j+k

X:=2e+3i4j+k

(2.3)

 

Calculate the norm of X.

Qalg > 

AlgebraNormX

30

(2.4)

 

Calculate the inverse of X and check the result.

Qalg > 

YAlgebraInverseX

Y:=115e110i+215j130k

(2.5)
Qalg > 

evalDGX.Y

e

(2.6)

 

Example 2.

Use AlgebraData to retrieve the structure equations for the octonions. Call the algebra Oalg.

ADAlgebraLibraryDataOctonions,Oalg

AD:=e12=e1,e1.e2=e2,e1.e3=e3,e1.e4=e4,e1.e5=e5,e1.e6=e6,e1.e7=e7,e1.e8=e8,e2.e1=e2,e22=e1,e2.e3=e4,e2.e4=e3,e2.e5=e6,e2.e6=e5,e2.e7=e8,e2.e8=e7,e3.e1=e3,e3.e2=e4,e32=e1,e3.e4=e2,e3.e5=e7,e3.e6=e8,e3.e7=e5,e3.e8=e6,e4.e1=e4,e4.e2=e3,e4.e3=e2,e42=e1,e4.e5=e8,e4.e6=e7,e4.e7=e6,e4.e8=e5,e5.e1=e5,e5.e2=e6,e5.e3=e7,e5.e4=e8,e52=e1,e5.e6=e2,e5.e7=e3,e5.e8=e4,e6.e1=e6,e6.e2=e5,e6.e3=e8,e6.e4=e7,e6.e5=e2,e62=e1,e6.e7=e4,e6.e8=e3,e7.e1=e7,e7.e2=e8,e7.e3=e5,e7.e4=e6,e7.e5=e3,e7.e6=e4,e72=e1,e7.e8=e2,e8.e1=e8,e8.e2=e7,e8.e3=e6,e8.e4=e5,e8.e5=e4,e8.e6=e3,e8.e7=e2,e82=e1

(2.7)

 

Initialize the algebra of octonions. We shall use the labelling e1, e2, e3, e4, e5, e6, e7 for the basis vectors with e1 being the identity.

DGsetupAD

algebra name: Oalg

(2.8)

 

Define an octonion X:

Qalg > 

X2e1+3e44e5+6e7

X:=2e1+3e44e5+6e7

(2.9)

 

Calculate the norm of X.

Qalg > 

AlgebraNormX

65

(2.10)

 

Calculate the inverse of X and check the result.

Qalg > 

YAlgebraInverseX

Y:=265e1365e4+465e5665e7

(2.11)
Qalg > 

evalDGX.Y

e1

(2.12)

See Also

DifferentialGeometry

Library

LieAlgebras

AlgebraData

DGconjugate

 


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