30 4. EXPONENTIAL AND LOGARITHM OF MATRICES

P

1

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.

log

cos sin

sin cos

D

0

0

;

log

0

@

e

a

0 0

0 e

b

0

0 0 e

c

1

A

D

0

@

a 0 0

0 b 0

0 0 c

1

A

;

and if A

m

D O for some m, then

log A D

m1

X

kD1

.1/

k1

k

.A I /

k

:

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

'

0

.0/ D

d

dt

'.t/

ˇ

ˇ

ˇ

ˇ

tD0

(4.14)

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/

T

D I

n

. By diﬀerentiating with the Leibnitz rule,

we obtain

0 D

d

dt

'.t/'.t/

T

ˇ

ˇ

ˇ

ˇ

tD0

D

d'.t/

dt

'.t/

T

C '.t/

d'.t/

T

dt

ˇ

ˇ

ˇ

ˇ

tD0

D A C A

T

;

where we put A D '

0

.0/ for short. is shows the Lie algebra of SO.n/ is the set of matrices A

with A

T

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

C

.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

T

D Ag.

(iii) e Lie algebra of the group of positive real numbers R

>0

is R. Note that the group law for

R

>0

is multiplicative, while that for R is additive.

(iv) e Lie algebra of C

is C,

(v) e Lie algebra of H

D fq 2 H j q ¤ 0g is H,

(vi) e Lie algebra of the group S

3

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

3

.

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