John Stillwell - Naive Lie Theory.pdf - Index of

142 7 The matrix logarithm
7.1.1 Supposing x = a 0 + a 1 y+a 2 y 2 + ··· (the function we call e y − 1), show that
y = (a 0 + a 1 y + a 2 y 2 + ···)
− 1 2 (a 0 + a 1 y + a 2 y 2 + ···) 2
+ 1 3 (a 0 + a 1 y + a 2 y 2 + ···) 3 ··· (*)
7.1.2 By equating the constant terms on both sides of (*), show that a 0 = 0.
7.1.3 By equating coefficients of y on both sides of (*), show that a 1 = 1.
7.1.4 By equating coefficients of y 2 on both sides of (*), show that a 2 = 1/2.
7.1.5 See whether you can go as far as Newton, who also found that a 3 = 1/6,
a 4 = 1/24, and a 5 = 1/120.
Newton then guessed that a n = 1/n! “by observing the analogy of the series.”
Unlike us, he did not have independent knowledge of the exponential function
ensuring that its coefficients follow the pattern observed in the first few.
As with exp, term-by-term differentiation and series manipulation give some
familiar formulas.
7.1.6 Prove that
dt d log(1 + At)=A(1 + At)−1 .
7.2 The exp function on the tangent space
For all the groups G we have seen so far it has been easy to find a general
form for tangent vectors A ′ (0) from the equation(s) defining the members
A of G. We can then check that all the matrices X of this form are mapped
into G by exp, and that e tX lies in G along with e X , in which case X is a
tangent vector to G at 1. Thus exp solves the problem of finding enough
smooth paths in G to give the whole tangent space T 1 (G)=g.
But if we are not given an equation defining the matrices A in G, we
may not be able to find tangent matrices in the form A ′ (0) in the first place,
so we need a different route to the tangent space. The log function looks
promising, because we can certainly get back into G by applying exp to a
value X of the log function, since exp inverts log.
However, it is not clear that log maps any part of G into T 1 (G), except
the single point 1 ∈ G. We need to make a closer study of the relation
between the limits that define tangent vectors and the definition of log.
This train of thought leads to the realization that G must be closed under
certain limits, and it prompts the following definition (foreshadowed in
Section 1.1) of the main concept in this book.