30 4. EXPONENTIAL AND LOGARITHM OF MATRICES
log.X/P , and then the computation of log.X / is reduced to the case of scalar-valued func-
tions and the case of triangular matrices, where inﬁnite series terminates.
0 0 e
a 0 0
0 b 0
0 0 c
and if A
D O for some m, then
log A D
.A I /
4.2 LIE ALGEBRA
e set so.3/ of skew-symmetric, that is, the transpose is its minus, 3 3 matrices is regarded as
Lie algebra of SO.3/ (see Chapter 1 as well).
Figure 4.1: Lie algebra as a tangent space.
In this book, we consider a Lie group to consist of matrices, say, a matrix group. e Lie
algebra of a Lie group is a linear approximation of a group at the origin. In general, the Lie algebra
of a matrix group G GL.N / is deﬁned to be the tangent space (see Figure 4.1) of G at the origin
of G. Equivalently, the Lie algebra is the collection of elements in M.N; R/ of the form
for any curve ' W R ! G with '.0/ D I .
4.2. LIE ALGEBRA 31
Sophus Lie (1862–1899)
Norwegian mathematician. He tried to control the continuous
symmetry in geometry and diﬀerential equations by introducing
its linearization. is idea is now regarded as a core of Lie eory.
Lie groups and Lie algebras are also named after him.
Lie was a close friend of F. Klein, and this communication
led to mutual inﬂuence (see [Stubhaug2002]).
For example, let us compute the Lie algebra of the Lie group SO.n/. For any curve ' W
R ! SO.n/, the image should satisfy '.t/'.t/
. By diﬀerentiating with the Leibnitz rule,
D A C A
where we put A D '
.0/ for short. is shows the Lie algebra of SO.n/ is the set of matrices A
D A. In general, the Lie algebra corresponding to a given Lie group is denoted by the
corresponding “mathfrak” letters; for example, the Lie algebra of SO.3/ is denoted by so.3/, that
of GL.n/ by gl.n/. We give several examples of Lie algebras.
(i) e Lie algebra of GL.n/ and GL
.n/ is gl.n/ D M.n; R/,
(ii) e Lie algebra of O.n/ and SO.n/ is so.n/ D fA 2 M.n; R/ j A
(iii) e Lie algebra of the group of positive real numbers R
is R. Note that the group law for
is multiplicative, while that for R is additive.
(iv) e Lie algebra of C
(v) e Lie algebra of H
D fq 2 H j q ¤ 0g is H,
(vi) e Lie algebra of the group S
of unit quaternions is the set Im H of imaginary quaternions.
(vii) e special linear group is deﬁned to be the set of volume-preserving linear maps;
SL.n; R/ D fA 2 GL.n; R/ j det A D 1g. Its Lie algebra is sl.n; R/ D fA 2 M.n; R/ j
trA D 0g, the set of traceless matrices.
(viii) e Lie algebra of the group of translations in 3D is R