Quaternions - Definitions

Helmut Knaust, Department of Mathematical Sciences, UTEP, El Paso TX 79968
hknaust@utep.edu
2/24/2012; last edits: 3/25/2014

A={{z,-w\[Conjugate]},{w,z\[Conjugate]}}

Here are the 4 basis elements:

ı=A/.{z->1,w->0}

i=A/.{z->0,w->1}

j=A/.{z->I,w->0}

k=A/.{z->0,w->I}

Hamilton's relations now hold:

{i.i==-ı,j.j==-ı,k.k==-ı,
i.j==k,j.k==i,k.i==j,
j.i==-k,k.j==-i,i.k==-j}

The quaternion a + b i +c j +d k as a matrix looks like this:

({a,b,c,d}.{ı,i,j,k})

With matrices, multiplication looks like this:

Row[{({a,b,c,d}.{ı,i,j,k})," · ",({e,f,g,h}.{ı,i,j,k})," = ",ComplexExpand[({a,b,c,d}.{ı,i,j,k}).({e,f,g,h}.{ı,i,j,k})]}]

Mathematica uses ‘Quaternion[a,b,c,d]’ to denote the quaternion a + b i +c j +d k, and ‘**’ to denote quaternion multiplication:

<<Quaternions`
Row[{Quaternion[a,b,c,d]," · ",Quaternion[e,f,g,h]," = ",Quaternion[a,b,c,d]**Quaternion[e,f,g,h]}]

Multiplication of quaternions in general is not commutative:

Row[{Quaternion[e,f,g,h]," · ",Quaternion[a,b,c,d]," = ",Quaternion[e,f,g,h]**Quaternion[a,b,c,d]}]

Quaternion[a,b,c,d]**Quaternion[a,-b,-c,-d]