Digital Signal Processing Reference
In-Depth Information
Thanks to their remarkable
algebraic
properties, and essentially their link with
one-parameter subgroups, matrix exponential and logarithms can be quite efficiently
numerically computed. In practice, we have used in this work the popular 'Scaling
and Squaring Method' [
35
] to compute numerically matrix exponentials, as well as
the 'Inverse Scaling and Squaring Method' [
18
] to compute matrix logarithms.
One Parameter Subgroups and Lie Group Exponential
Let us now define the general group exponential and logarithm in Lie groups. These
properties are very similar to those of the matrix exponential and logarithm, which
are a particular case of such mappings. One should note that this particular case is
actually quite general, since most classical Lie groups can be looked upon as matrix
Lie groups anyway [
33
].
The flow
γ
x
(
X
t
)
of a left-invariant vector field
=
DL x
starting from e exists for
all times. Its tangent vector is
γ
x
(
t
)
=
DL
γ
x
(
t
)
x
by definition of the flow. Now fix
s
∈ R
and observe that the two curves
γ
x
(
s
+
t
)
and
γ
x
(
s
)
·
γ
x
(
t
)
are going through
point
γ
x
(
)
=
0 with the same tangent vector. By the uniqueness of the
flow, they are the same and
γ
x
is a one parameter subgroup, i.e. a group morphism
from
s
at time
t
(G,
,
·
)
(
R
,
,
+
)
e
to
0
:
γ
x
(
s
+
t
)
=
γ
x
(
s
)
·
γ
x
(
t
)
=
γ
x
(
t
+
s
)
=
γ
x
(
t
)
·
γ
x
(
s
).
The group exponential is defined from these one-parameter subgroups with Exp
(
x
)
=
γ
x
(
1
)
.
Definition 7.1
be a Lie group and let
x
be an element of the Lie Algebra
g
.
The group exponential of
x
, denoted Exp
Let
G
(
x
)
, is given by the value at time 1 of the
unique function
γ
x
(
t
)
defined by the ordinary differential equation (ODE)
γ
x
(
t
)
=
DL
γ
x
(
t
)
x
with initial condition
γ
x
(
0
)
=
e.
Very much like the exponential map associated to a Riemannian metric, the group
exponential is diffeomorphic locally around 0. More precisely, since the exponential
is a smooth mapping, the fact that its differential map is invertible at e allows for the
use of the 'Inverse Function Theorem', which guarantees that it is a diffeomorphism
from some open neighborhood of 0 to an open neighborhood of Exp
(
)
=
0
e[
58
,
Proposition 1.3, p. 13].
Theorem 7.1
The group exponential is a diffeomorphism from an open neighbor-
hood of
0
in
g
to an open neighborhood of
e
in
G
, and its differential map at
0
is the
identity.
This theorem implies that one can define
without ambiguity
a logarithm in an
open neighborhood of e: for every
g
in this open neighborhood, there exists a unique
x
in the open neighborhood of 0 in
g
, such that
g
=
Exp
(
x
)
. In the following,
we will write
x
for this logarithm, which is the (abstract) equivalent
of the (matrix)
principal
logarithm. The absence of an inverse function theorem in
=
Log
(g)
