Quaternionic Computing

Furthermore,theusualvectorinnerproducthastherequiredproperties(i.e.itisnormde ning),andaproperHilbertspacecanbede nedonanyquaternioniclinearspace.

Itisalsopossibletocomplexifythequaternions,thisis,torepresentthemintermsofcomplexnumbersonly.Letα beanarbitraryquaternion,thenwede neitscomplexandweirdpartsas

Co( α) a0+a1i

Wd( α) a2+a3i.

Wecanthendecomposeα initscomplexandweirdpartasfollows:

α =a0+a1i+a2j+a3k

=(a0+a1i)+(a2+a3i)j

=Co( α)+Wd( α)j

Thisequationallowsustoderivemultiplicationrules,similartothoseofEquation11

)=Co( ) Wd( )Co( αβα)Co(βα)Wd (β

)=Co( )+Wd( )Wd( αβα)Wd(βα)Co (β(41)(38)(39)(40)

wherewede neCo ( α) [Co( α)] ,andsimilarlyfortheweirdpartWd ( α) [Wd( α)] .Itisinterestingtonotehowthenon-commutativityofquaternionsismadeapparentbythefactthat ,unliketheirequivalentfor andβneitheridentityinEquation41issymmetricwithrespecttoα

complexnumbers(Equation11),becauseingeneralCo ( α)=Co( α)andWd ( α)=Wd( α).WecanalsorewriteEquation37forthemodulusas

|α |=

|2|α |2+|β

uptoanarbitraryquaternionicphasefactor.Indeed,wehavethat

Φ≡Φ′ |Φ =η |Φ′ ,where|η |=1.(44)(45)