John Stillwell - Naive Lie Theory.pdf - Index of

144 7 The matrix logarithm
Taking exp of equation (**), we get
e A′ (0) = e lim n→∞ nlogA(1/n)
nlog A(1/n)
= lim e because
n→∞
= lim
(
e log A(1/n)) n
n→∞
= lim
n→∞
A(1/n) n
exp is continuous
because e A+B = e A e B when AB = BA
because exp is the inverse of log.
Now A(1/n) ∈ G by assumption, so A(1/n) n ∈ G because G is closed under
products. We therefore have a convergent sequence of members of G, and
its limit e A′ (0) is nonsingular because it has inverse e −A′ (0) .Soe A′ (0) ∈ G,
by the closure of G under nonsingular limits.
In other words, exp maps the tangent space T 1 (G)=g into G. □
The proof in the opposite direction, from G into T 1 (G), is more subtle.
It requires a deeper study of limits, which we undertake in the next section.
Exercises
7.2.1 Deduce from exponentiation of tangent vectors that
T 1 (G)={X : e tX ∈ G for all t ∈ R}.
The property T 1 (G)={X : e tX ∈ G for all t ∈ R} is used as a definition of T 1 (G)
by some authors, for example Hall [2003]. It has the advantage of making it clear
that exp maps T 1 (G) into G. On the other hand, with this definition, we have to
check that T 1 (G) is a vector space.
7.2.2 Given X as the tangent vector to e tX ,andY as the tangent vector to e tY ,
show that X +Y is the tangent vector to A(t)=e tX e tY .
7.2.3 Similarly, show that if X is a tangent vector then so is rX for any r ∈ R.
The formula A ′ (0)=lim n→∞ nlogA(1/n) that emerges in the proof above can
actually be used in two directions. It can be used to prove that exp maps T 1 (G)
into G when combined with the fact that G is closed under products (and hence
under nth powers). And it can be used to prove that log maps (a neighborhood of
1 in) G into T 1 (G) when combined with the fact that G is closed under nth roots.
Unfortunately, proving closure under nth roots is as hard as proving that log
maps into T 1 (G), so we need a different approach to the latter theorem. Nevertheless,
it is interesting to see how nth roots are related to the behavior of the log
function, so we develop the relationship in the following exercises.