John Stillwell - Naive Lie Theory.pdf - Index of

148 7 The matrix logarithm
The log of a neighborhood of 1. For any matrix Lie group G there is a
neighborhood N δ (1) mapped into T 1 (G) by log.
Proof. Suppose on the contrary that no N δ (1) is mapped into T 1 (G) by log.
Then we can find A 1 ,A 2 ,A 3 ,...∈ G with A m → 1 as m → ∞, and with each
logA m ∉ T 1 (G).
Of course, G is contained in some M n (C). So each log A m is in M n (C)
and we can write
log A m = X m +Y m ,
where X m is the component of logA m in T 1 (G) and Y m ≠ 0 is the component
in T 1 (G) ⊥ , the orthogonal complement of T 1 (G) in M n (C). We note that
X m ,Y m → 0 as m → ∞ because A m → 1 and log is continuous.
Next we consider the matrices Y m /|Y m |∈T 1 (G) ⊥ . These all have absolute
value 1, so they lie on the sphere S of radius 1 and center 0 in
M n (C). It follows from the boundedness of S that the sequence 〈Y m /|Y m |〉
has a convergent subsequence, and the limit Y of this subsequence is also
a vector in T 1 (G) ⊥ of length 1. In particular, Y ∉ T 1 (G).
Taking the subsequence with limit Y in place of the original sequence
we have
Y m
lim
m→∞ |Y m | = Y.
Finally, we consider the sequence of terms
T m = e −X m
A m .
Each T m ∈ G because −X m ∈ T 1 (G); hence e −X m
∈ G by the exponentiation
of tangent vectors in Section 7.2, and A m ∈ G by hypothesis. On the other
hand, A m = e X m+Y m
by the inverse property of log, so
T m = e −X m
e X m+Y m
(
= 1 − X m + X m
2 ···)(
2! + 1 + X m +Y m + (X m +Y m ) 2 )
+ ···
2!
= 1 +Y m + higher-order terms.
Admittedly, these higher-order terms include X 2 m, and other powers of X m ,
that are not necessarily small in comparison with Y m . However, these powers
of X m are those in
1 = e −X m
e X m
,