Quaternions - Definitions

Helmut Knaust, Department of Mathematical Sciences, UTEP, El Paso TX 79968
hknaust@utep.edu
2/24/2012; last edits: 03/19/2024

We  let ℍ be the set of all matrices of the form (({{z, -(Conjugate[w])}, {w, (Conjugate[z])}})) with z,w ϵ ℂ. (* stands for conjugation.)

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}

Every quaternion in ℍ has the form a + b i +c j +d k with a,b,c,d ϵ ℝ. As a matrix it looks like this:

({a,b,c,d}.{ı,i,j,k})

ℍ is a four-dimensional  ℝ-vector space.

It is not a ℂ-vector  space:

A
I A
(I z)\[Conjugate]

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]}]

Every element in ℍ other than 0 has a multiplicative inverse:

w=Quaternion[a,b,c,d]

Conjugate[w]

w**Conjugate[w]

Here  is  an example:

Quaternion[1,2,-3,4]**(1/30Quaternion[1,-2,3,-4])

FromQuaternion[%]