Quaternionic Computing

Proof.The rststepistoobtainasimplematrixmultiplicationruleformatrices,usingtheoperatorsReandIm.Forarbitrarycomplexnumbersαandβ,wehavethat

Re(αβ)=Re(α)Re(β) Im(α)Im(β)

Im(αβ)=Re(α)Im(β)+Im(α)Re(β)(11)

Sincetheserulesholdfortheproductsofalloftheirentries,itistheneasytoseethatthissamemultiplicationrulewillalsoholdforcomplexmatrices.Inotherwords,wecansubstituteαandβinEquation11withanytwoarbitrarycomplexmatricesAandBwhicharemultipliable,toget

Re(AB)=Re(A)Re(B) Im(A)Im(B)

Im(AB)=Re(A)Im(B)+Im(A)Re(B)

Wearenowequippedtoverifyourclaim

h(A)h(B)=(T A)(T B) Re(A)Im(B)= Im(A)Re(B) Re(A)Re(B) Im(A)Im(B)= Im(A)Re(B) Re(A)Im(B) Im(AB)

Re(AB)

=T AB=h(AB)(13)(12)

Finally,wewanttoshowthatGN SO(2N).ThisisequivalenttoshowingthatalltheimagesO=h(U)areorthonormal,i.e.thatOt=O 1.SincebyLemma1hisagrouphomomorphism,itmapsinverseelementsintoinverseelements,i.e.h(U 1)=h(U) 1.SinceUisunitary,wehavethat

O 1=h(U) 1=h(U 1)=h(U )(14)

whilethefollowinglemmawillgiveusanexpressionforOt.

Lemma2.LetAbeanarbitraryN×Ncomplexmatrix,thenh(A )=h(A)t.

Proof.Byde nitionofhandbytranspositionrulesofblockmatrices,wehave

h(A)t=(T A)t

Re(A)= Im(A)

Im(A)t

Re(A)t

=Re(A )

Im(A )