John Stillwell - Naive Lie Theory.pdf - Index of

86 4 The exponential map
Definition. The exponential of any n × n matrix A is given by the series
e A = 1 + A 1! + A2
2! + A3
3! + ···.
The matrix exponential function is a generalization of the complex and
quaternion exponential functions. We already know that each complex
number z = a + bi can be represented by the 2 × 2 real matrix
( ) a −b
Z = ,
b a
and it is easy to check that e z is represented by e Z . We defined the quaternion
q = a + bi + cj + dk to be the 2 × 2 complex matrix
( )
a + di −b + ci
Q =
,
b + ci a − di
so the exponential of a quaternion matrix may be represented by the exponential
of a complex matrix.
From now on we will often denote the exponential function simply by
exp, regardless of the type of objects being exponentiated.
Exercises
The version of the Cauchy–Schwarz inequality used to prove the submultiplicative
property is the real inner product inequality |u · v|≤|u||v|,where
u =(|a i1 |,|a i2 |,...,|a in |) and v = ( |b j1 |,|b j2 |,...,|b jn | ) .
It is probably a good idea for me to review this form of Cauchy–Schwarz, since
some readers may not have seen it.
The proof depends on the fact that w · w = |w| 2 ≥ 0 for any real vector w.
4.5.1 Show that 0 ≤ (u + xv) · (u + xv)=|u| 2 + 2(u · v)x + x 2 |v| 2 = q(x), forany
real vectors u, v and real number x.
4.5.2 Use the positivity of the quadratic function q(x) found in Exercise 4.5.1 to
deduce that
(2u · v) 2 − 4|u| 2 |v| 2 ≤ 0,
that is, |u · v|≤|u||v|.