Number Theory: An Introduction to Mathematics

52 I The Expanding Universe of Numbers

If we write

x=ξ 1 i+ξ 2 j+ξ 3 k,

then

Tx=y=η 1 i+η 2 j+η 3 k,

whereημ=

∑ 3

v= 1 βμvξvfor someβμv∈R.Since

η^21 +η^22 +η^23 =ξ 12 +ξ 22 +ξ 32 ,

the matrixV=(βμv)isorthogonal:V−^1 =Vt.
Thus with every quaternionuwith norm 1 there is associated a 3×3 orthogonal
matrixV=(βμv). Explicitly, if

u=α 0 +α 1 i+α 2 j+α 3 k,

where

α 02 +α^21 +α^22 +α^23 = 1 ,

then

β 11 =α^20 +α^21 −α^22 −α 32 ,β 12 = 2 (α 1 α 2 −α 0 α 3 ), β 13 = 2 (α 1 α 3 +α 0 α 2 ),

β 21 = 2 (α 1 α 2 +α 0 α 3 ), β 22 =α^20 −α^21 +α^22 −α^23 ,β 23 = 2 (α 2 α 3 −α 0 α 1 ),

β 31 = 2 (α 1 α 3 −α 0 α 2 ), β 32 = 2 (α 2 α 3 +α 0 α 1 ), β 33 =α 02 −α^21 −α^22 +α^23.

This parametrization of orthogonal transformations was first discovered by Euler(1770).
We now consider the dependence ofVonu, and consequently writeV(u)in place
ofV.Since

u 1 u 2 x(u 1 u 2 )−^1 =u 1 (u 2 xu− 21 )u− 11 ,

we have

V(u 1 u 2 )=V(u 1 )V(u 2 ).

Thus the mapu→ V(u)is a ‘homomorphism’ of the multiplicative group of all
quaternions of norm 1 into the group of all 3×3 real orthogonal matrices. In particu-
lar,V(u ̄)=V(u)−^1.
We show next that two quaternionsu 1 ,u 2 of norm 1 yield the same orthogonal
matrix if and only ifu 2 =±u 1 .Putu=u− 21 u 1 .Thenu 1 xu− 11 =u 2 xu− 21 if and only
ifux=xu. This holds for every pure quaternionxif and only ifuisreal,i.e.ifand
only ifu=±1, sincen(u)=1.
The question arises whether all 3×3 orthogonal matrices may be represented in
the above way. It follows readilyfrom the preceding formulas forβμvthat the orthog-
onal matrix−Icannot be so represented. Consequently, if an orthogonal matrixVis