20 CHAPTER 1. THE MATRIX EXPONENTIAL; SOME MATRIX LIE GROUPS
special orthogonal group SO(n), and the group of affine rigid motions SE(n), and their Lie
algebras gl(n, R) (all matrices), sl(n, R) (matrices with null trace), o(n), and so(n) (skew
symmetric matrices). Lie groups are at the same time, groups, topological spaces, and
manifolds, so we will also have to introduce the crucial notion of a manifold.
The inventors of Lie groups and Lie algebras (starting with Lie!) regarded Lie groups as
groups of symmetries of various topological or geometric objects. Lie algebras were viewed
as the “infinitesimal transformations” associated with the symmetries in the Lie group. For
example, the group SO(n) of rotations is the group of orientation-preserving isometries of
the Euclidean space E
n
. The Lie algebra so(n, R) consisting of real skew symmetric n × n
matrices is the corresponding set of infinitesimal rotations. The geometric link between a Lie
group and its Lie algebra is the fact that the Lie algebra can be viewed as the tangent space
to the Lie group at the identity. There is a map from the tangent space to the Lie group,
called the exponential map. The Lie algebra can be considered as a linearization of the Lie
group (near the identity element), and the exponential map provides the “delinearization,”
i.e., it takes us back to the Lie group. These concepts have a concrete realization in the
case of groups of matrices and, for this reason, we begin by studying the behavior of the
exponential maps on matrices.
We begin by defining the exponential map on matrices and proving some of its properties.
The exponential map allows us to “linearize” certain algebraic properties of matrices. It also
plays a crucial role in the theory of linear differential equations with constant coefficients.
But most of all, as we mentioned earlier, it is a stepping stone to Lie groups and Lie algebras.
On the way to Lie algebras, we derive the classical “Rodrigues-like” formulae for rotations
and for rigid motions in R
2
and R
3
. We give an elementary proof that the exponential map
is surjective for both SO(n) and SE(n), not using any topology, just certain normal forms
for matrices (see Gallier [76], Chapters 12 and 13).
Chapter 4 gives an introduction to manifolds, Lie groups and Lie algebras. Rather than
defining abstract manifolds in terms of charts, atlases, etc., we consider the special case of
embedded submanifolds of R
N
. This approach has the pedagogical advantage of being more
concrete since it uses parametrizations of subsets of R
N
, which should be familiar to the
reader in the case of curves and surfaces. The general definition of a manifold will be given
in Chapter 7.
Also, rather than defining Lie groups in full generality, we define linear Lie groups us-
ing the famous result of Cartan (apparently actually due to Von Neumann) that a closed
subgroup of GL(n, R) is a manifold, and thus a Lie group. This way, Lie algebras can be
“computed” using tangent vectors to curves of the form t 7→ A(t), where A(t) is a matrix.
This section is inspired from Artin [11], Chevalley [44], Marsden and Ratiu [125], Curtis [49],
Howe [99], and Sattinger and Weaver [159].
Given an n×n (real or complex) matrix A = (a
i j
), we would like to define the exponential
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