 27
C H A P T E R 4
Exponential and Logarithm of
Matrices
Geometric transformations that we have described give a basic mathematical framework for ge-
ometric operations in computer graphics, such as rotation, shear, translation, and their compo-
sitions. Each aﬃne transformation is then represented by a 4 4-homogeneous matrix (2.43)
with usual operations: addition, scalar product, and product. While the product means the com-
position of the transformations, geometric meanings of addition and scalar product are not triv-
ial. We often want to have a geometrically meaningful weighted sum (linear combination) of
transformations, which is not an easy task. ese kinds of practical demands therefore have in-
spired graphics researchers to explore new mathematical concepts and/or tools. Many works have
been conducted in this direction, including skinning [Chaudhry2010] [Lewis2000], cage-based
deformation [Nieto2013], motion analysis and compression ([Alexa2002], [Tournier2009], for
instance).
In previous chapters we have been redescribing the geometric transformations through the
mathematical concepts associated with groups, especially with the Lie group. We should then
note that the mathematical viewpoint gives us a broader and more comprehensive scope of the
various geometric transformations.
is chapter focuses a bit more on the Lie theoretic aspect of this scope. We introduce Lie
algebra that associates the Lie group of matrices as a motion group. As will be demonstrated, the
Lie algebra gives linear approximation of the Lie group, which allows us to use a powerful linear
interpolation scheme in making dynamic motion and deformation.
4.1 EXPONENTIAL: DEFINITIONS AND BASIC
PROPERTIES
We ﬁrst consider a square matrix A, which is implicitly considered an element of a Lie group of
matrices. e exponential of A is then deﬁned as
exp.A/ D
1
X
nD0
1
A
n
D I C A C
1
2
A
2
C
1
6
A
3
C ; (4.1) 28 4. EXPONENTIAL AND LOGARITHM OF MATRICES
where A
0
D I is the identity matrix. We’ll refer to (4.1) as the matrix exponential, for short. is
is motivated by Taylor expansion of the usual exponential function
e
x
D
1
X
nD0
1
x
n
D 1 C x C
1
2
x
2
C
1
x
3
C : (4.2)
e series exp.A/ converges for an arbitrary A rapidly, as does the usual exponential function.
However, this inﬁnite series expression is not so eﬃcient for actual numerical computations. For
a computation, we can use several useful properties: for diagonal matrices we have
exp
0
@
a 0 0
0 b 0
0 0 c
1
A
D
0
@
e
a
0 0
0 e
b
0
0 0 e
c
1
A
; (4.3)
and a rotation
exp
0
0
D
cos sin
sin cos
: (4.4)
We also see that the exponential image of a strictly upper-triangular matrix terminates into a
ﬁnite sum. For example, for l 2 R
n
, we have
exp
O l
0 0
D
I
n
l
0 1
: (4.5)
Slightly more generally, for a strictly upper triangular matrix A of size n, we have A
n
D O and
therefore the inﬁnite series (4.1) terminates to a ﬁnite sum expression
exp.A/ D I C A C
1
2
A
2
C C
1
.n 1/Š
A
n1
: (4.6)
Signiﬁcantly, we can understand Rodrigues’s rotation formula (2.15) by using the matrix
exponential. Every 3D rotation is expressed by
R D exp.A/ D exp
0
@
0 u
3
u
2
u
3
0 u
1
u
2
u
1
0
1
A
D I
3
C
sin
j
u
j
j
u
j
A C
1 cos
j
u
j
j
u
j
2
A
2
; (4.7)
where
j
u
j
D
q
u
2
1
C u
2
2
C u
2
3
is the norm of a vector u D .u
1
; u
2
; u
3
/ 2 R
3
. We also see that
j
u
j
2
D
1
2
tr.AA
T
/ D
1
2
tr.A
2
/. e matrix R shows the rotation around the axis through u, and
with angle
j
u
j
. In particular, if
j
u
j
2 2Z then R D I
3
, the identity matrix.
Coming back to the general situation, we always have the exponential law
exp..s C t/A/ D exp.sA/ exp.tA/ for all s; t 2 R; A 2 M.n; R/: (4.8)

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