John Stillwell - Naive Lie Theory.pdf - Index of

96 5 The tangent space
The matrices X satisfying X + X T = 0 are called skew-symmetric, because
the reflection of each entry in the diagonal is its negative. That is,
x ji = −x ij . In particular, all the diagonal elements of a skew-symmetric
matrix are 0. Matrices X satisfying X +X T = 0 are called skew-Hermitian.
Their entries satisfy x ji = −x ij and their diagonal elements are pure imaginary.
It turns out that all skew-symmetric n × n real matrices are tangent,
not only to O(n), but also to SO(n) at 1. To prove this we use the matrix
exponential function from Section 4.5, showing that e X ∈ SO(n) for any
skew-symmetric X, in which case X is tangent to the smooth path e tX in
SO(n).
Exercises
To appreciate why smooth paths are better than mere paths, consider the following
example.
5.1.1 Interpret the paths B(t) and C(t) above as paths on the unit circle, say for
−π/2 ≤ t ≤ π/2.
5.1.2 If B(t) or C(t) is interpreted as the position of a point at time t, how does
the motion described by B(t) differ from the motion described by C(t)?
5.2 The tangent space of SO(n)
In this section we return to the addition formula of the exponential function
e A+B = e A e B when AB = BA,
which was previously set as a series of exercises in Section 4.1. This formula
can be proved by observing the nature of the calculation involved,
without actually doing any calculation. The argument goes as follows.
According to the definition of the exponential function, we want to
prove that
(
1 + A + B
1!
(A + B)n
+ ···+
n!
=
(1 + A 1!
)
+ ···
···)(
An
+ ···+
n! + 1 + B ···)
Bn
+ ···+
1! n! + .
This could be done by expanding both sides and showing that the coefficient
of A l B m is the same on both sides. But if AB = BA the calculation