John Stillwell - Naive Lie Theory.pdf - Index of

1.4 Consequences of multiplicative absolute value 11
1.3.3 Check that q is the result of taking the complex conjugate of each entry in
q T , and hence show that q 1 q 2 = q 2 q 1 for any quaternions q 1 and q 2 .
1.3.4 Also check that qq = |q| 2 .
Cayley’s matrix representation makes it easy (in principle) to derive an amazing
algebraic identity.
1.3.5 Show that the multiplicative property of determinants gives the complex
two-square identity (discovered by Gauss around 1820)
(|α 1 | 2 + |β 1 | 2 )(|α 2 | 2 + |β 2 | 2 )=|α 1 α 2 − β 1 β 2 | 2 + |α 1 β 2 + β 1 α 2 | 2 .
1.3.6 Show that the multiplicative property of determinants gives the real foursquare
identity
(a 2 1 + b2 1 + c2 1 + d2 1 )(a2 2 + b2 2 + c2 2 + d2 2 )= (a 1a 2 − b 1 b 2 − c 1 c 2 − d 1 d 2 ) 2
+(a 1 b 2 + b 1 a 2 + c 1 d 2 − d 1 c 2 ) 2
+(a 1 c 2 − b 1 d 2 + c 1 a 2 + d 1 b 2 ) 2
+(a 1 d 2 + b 1 c 2 − c 1 b 2 + d 1 a 2 ) 2 .
This identity was discovered by Euler in 1748, nearly 100 years before the discovery
of quaternions! Like Diophantus, he was interested in the case of integer
squares, in which case the identity says that
(a sum of four squares)× (a sum of four squares) = (a sum of four squares).
This was the first step toward proving the theorem that every positive integer is
the sum of four integer squares. The proof was completed by Lagrange in 1770.
1.3.7 Express 97 and 99 as sums of four squares.
1.3.8 Using Exercise 1.3.6, or otherwise, express 97×99 as a sum of four squares.
1.4 Consequences of multiplicative absolute value
The multiplicative absolute value, for both complex numbers and quaternions,
first appeared in number theory as a property of sums of squares. It
was noticed only later that it has geometric implications, relating multiplication
to rigid motions of R 2 , R 3 ,andR 4 . Suppose first that u is a complex
number of absolute value 1. Without any computation with cosθ and sinθ,
we can see that multiplication of C = R 2 by u is a rotation of the plane as
follows.