From Discrete to Continuous Symmetry
In Geometry of Symmetry, we explored the
dihedral group
\(D_n\), the finite group of symmetries of a regular \(n\)-gon. The same page previewed two
continuous groups, \(SO(3)\) and \(SE(3)\), that govern rotation and rigid body motion in
three-dimensional space.
There, we observed these continuous groups visually and intuitively. The group \(SO(3)\) allows
rotations by any angle, and \(SE(3)\) combines such rotations with arbitrary
translations. We noted that the passage from \(D_n\) to \(SO(3)\) is a shift from
discrete to continuous symmetry. We did not, however, make this precise.
The present page fulfills this task. We formalize continuous symmetry groups as
matrix Lie groups and develop the matrix exponential, the
fundamental tool that connects infinitesimal symmetries to finite transformations. The key
question driving this development is: what does it mean for a group to be "continuous"?
Topological Groups
The answer lies in combining two structures we have studied separately: the algebraic structure
of a group and the topological structure of a space with a notion of continuity. A group whose
operations respect topology is called a topological group.
Definition: Topological Group
A topological group is a group \(G\) equipped with a topology such that
the following two maps are continuous:
-
Multiplication:
\(\mu : G \times G \to G\), \(\mu(g, h) = gh\), where \(G \times G\) carries the
product topology.
-
Inversion:
\(\iota : G \to G\), \(\iota(g) = g^{-1}\).
The continuity of multiplication means that if \(g\) and \(h\) are "close" to \(g_0\) and
\(h_0\) respectively, then \(gh\) is close to \(g_0 h_0\). Continuity of inversion ensures that
nearby elements have nearby inverses. Together, they guarantee that the algebraic operations
are compatible with the topology. The group structure does not "tear" the underlying space.
Examples:
(a) \((\mathbb{R}, +)\) with the standard (Euclidean) topology is a
topological group. Addition \((x, y) \mapsto x + y\) and negation \(x \mapsto -x\)
are both continuous.
(b) \((\mathbb{R} \setminus \{0\}, \cdot)\) with the subspace topology
inherited from \(\mathbb{R}\) is a topological group. Multiplication and the map
\(x \mapsto 1/x\) are continuous on \(\mathbb{R} \setminus \{0\}\).
(c) The circle group
\(S^1 = \{z \in \mathbb{C} : |z| = 1\}\) with the subspace topology from \(\mathbb{C}\) is
a topological group under complex multiplication. Multiplication and inversion
\(z \mapsto \bar{z}\) are continuous. This is the simplest nontrivial example of a compact,
connected topological group.
Toward Lie Groups
A Lie group is a group that is simultaneously a smooth manifold,
with smooth (infinitely differentiable) group operations. Since
smooth manifolds have not
yet been formally defined in this curriculum, we take a concrete approach that is both fully
rigorous and historically prior to the
abstract definition.
Our strategy is to define matrix Lie groups as closed subgroups of the general
linear group \(GL(n, \mathbb{C})\). Cartan's Closed Subgroup Theorem, stated in the next
section, guarantees that every such closed subgroup is automatically a smooth manifold with
smooth group operations. In other words, closedness alone buys us all the smoothness we need.
This approach mirrors a recurring pattern. Just as
Natural Gradient Descent
previewed Riemannian geometry before manifolds were formally available, we now study the most
important Lie groups concretely before the general definition arrives. Every matrix Lie group
we define here will be a Lie group in the abstract sense. We lose nothing by starting with
matrices, and we gain the ability to compute.
Matrix Lie Groups
We now define the arena in which all our groups will live. The space
\(M_n(\mathbb{F})\) of \(n \times n\) matrices over \(\mathbb{F}\) (where
\(\mathbb{F} = \mathbb{R}\) or \(\mathbb{C}\)) is a finite-dimensional vector space
isomorphic to \(\mathbb{F}^{n^2}\), and we equip it with the topology induced by any
norm (all norms on a finite-dimensional space are equivalent). Within this space sits the
group of invertible matrices.
Definition: General Linear Group
The general linear group is
\[
GL(n, \mathbb{F}) = \{ A \in M_n(\mathbb{F}) : \det(A) \neq 0 \}.
\]
Since the determinant \(\det : M_n(\mathbb{F}) \to \mathbb{F}\) is a polynomial in the
matrix entries (hence continuous), and \(\{0\}\) is closed in \(\mathbb{F}\), the
complement \(GL(n, \mathbb{F}) = \det^{-1}(\mathbb{F} \setminus \{0\})\) is an
open subset of \(M_n(\mathbb{F})\).
Matrix multiplication is polynomial in the entries, hence continuous. Matrix
inversion is given by the
adjugate formula
\(A^{-1} = \operatorname{adj}(A)/\det(A)\). Each entry of \(A^{-1}\) is
therefore a rational function of the entries of \(A\) with denominator
\(\det(A) \neq 0\), hence continuous on \(GL(n, \mathbb{F})\). Therefore
\(GL(n, \mathbb{F})\) is a topological group.
The general linear group is the "universe" of matrix groups. Every matrix Lie group will
be a subgroup of \(GL(n, \mathbb{C})\) (or \(GL(n, \mathbb{R})\)) satisfying one additional
condition: closedness.
Definition: Matrix Lie Group
A matrix Lie group is a subgroup \(G \leq GL(n, \mathbb{C})\) with the
following closedness property. If \(A_1, A_2, A_3, \ldots \in G\) and \(A_k \to A\) in
\(M_n(\mathbb{C})\), then either \(A \in G\) or \(A \notin GL(n, \mathbb{C})\).
Equivalently, \(G\) is a closed subset of \(GL(n, \mathbb{C})\) with
respect to the
subspace topology
inherited from \(M_n(\mathbb{C})\).
The phrasing "either \(A \in G\) or \(A \notin GL(n, \mathbb{C})\)" can appear puzzling at
first. Its meaning is the following.
As long as the limit matrix remains invertible, it must belong to \(G\). The only way
a sequence in \(G\) can converge to a matrix outside \(G\) is by "escaping" \(GL(n)\)
entirely, that is, by having its determinant tend to zero.
For example, consider \(SL(n, \mathbb{R})\). If \(A_k \in SL(n, \mathbb{R})\) and \(A_k \to A\)
with \(\det(A) \neq 0\), then by continuity of the determinant,
\(\det(A) = \lim \det(A_k) = 1\), so \(A \in SL(n, \mathbb{R})\), and the group is therefore
closed. In contrast, take the sequence \(A_k = \frac{1}{k}I\) in \(GL(n, \mathbb{R})\). It has
\(\det(A_k) = k^{-n} \to 0\), so its limit \(A = 0\) is not even in \(GL(n)\). Such an escape
from invertibility does not violate closedness. Only a limit that remains invertible but falls
outside the group would.
The closedness condition is mild. It is the only topological condition we impose,
yet it has far-reaching consequences, as Cartan's theorem below will show. Intuitively,
closedness prevents the group from having "holes" or "missing boundary points" that would
destroy its manifold structure.
The Classical Groups
We now introduce the classical matrix Lie groups, the groups that appear throughout
mathematics, physics, and computer science. For each group, we verify that it is indeed a
matrix Lie group by checking that it is a closed subgroup of the ambient general linear group.
Several of the groups below sit inside \(GL(n, \mathbb{R})\), and closedness there is enough.
Indeed, \(M_n(\mathbb{R})\) is closed in \(M_n(\mathbb{C})\), so
\(GL(n, \mathbb{R}) = GL(n, \mathbb{C}) \cap M_n(\mathbb{R})\) is closed in
\(GL(n, \mathbb{C})\), and a subgroup closed in \(GL(n, \mathbb{R})\) is then closed in
\(GL(n, \mathbb{C})\) as well.
Definition: Special Linear Group
The special linear group is
\[
SL(n, \mathbb{R}) = \{ A \in GL(n, \mathbb{R}) : \det(A) = 1 \}.
\]
This is the group of volume- and orientation-preserving linear transformations.
Proof that \(SL(n, \mathbb{R})\) is a matrix Lie group:
The determinant map \(\det : GL(n, \mathbb{R}) \to \mathbb{R} \setminus \{0\}\)
is a continuous
group homomorphism
(with \(\mathbb{R} \setminus \{0\}\) under multiplication). Its
kernel
is \(SL(n, \mathbb{R}) = \det^{-1}(\{1\})\). Since \(\{1\}\) is closed in
\(\mathbb{R} \setminus \{0\}\) and \(\det\) is continuous, \(SL(n, \mathbb{R})\)
is a closed subgroup of \(GL(n, \mathbb{R})\). The dimension of
\(SL(n, \mathbb{R})\) is \(n^2 - 1\) (the single constraint \(\det(A) = 1\)
removes one degree of freedom).
Definition: Orthogonal Group
The orthogonal group is
\[
O(n) = \{ A \in GL(n, \mathbb{R}) : A^\top A = I \}.
\]
Equivalently, \(O(n)\) consists of the linear transformations that preserve the
Euclidean inner product: \(\langle Ax, Ay \rangle = \langle x, y \rangle\) for all
\(x, y \in \mathbb{R}^n\).
Recall that these are precisely the
orthogonal matrices
\(A\) satisfying \(A^\top A = I\), the norm-preserving transformations studied earlier.
Proof that \(O(n)\) is a closed subgroup of \(GL(n, \mathbb{R})\):
Consider the map \(\Phi : M_n(\mathbb{R}) \to M_n(\mathbb{R})\) defined by
\(\Phi(A) = A^\top A\). This map is continuous (it is polynomial in the entries), and
\(O(n) = \Phi^{-1}(\{I\})\). Since \(\{I\}\) is a closed set in \(M_n(\mathbb{R})\),
the preimage \(O(n)\) is closed. Furthermore, if \(A^\top A = I\), then
\(\det(A)^2 = \det(A^\top A) = 1\), so \(\det(A) = \pm 1 \neq 0\), confirming
\(O(n) \subset GL(n, \mathbb{R})\).
Proof that \(O(n)\) is compact:
Bounded. For any \(A \in O(n)\), the
Frobenius norm
satisfies \(\|A\|_F^2 = \mathrm{tr}(A^\top A) = \mathrm{tr}(I) = n\). Hence every element
of \(O(n)\) has the same Frobenius norm \(\sqrt{n}\), so \(O(n)\) is bounded in
\(M_n(\mathbb{R}) \cong \mathbb{R}^{n^2}\).
Closed. We proved this above.
By the
Heine-Borel theorem,
a subset of \(\mathbb{R}^{n^2}\) is compact if and only if it is closed and bounded.
Therefore \(O(n)\) is compact.
The condition \(\det(A) = \pm 1\) for \(A \in O(n)\) splits \(O(n)\) into two disjoint pieces:
the matrices with \(\det(A) = +1\) (proper rotations) and those with \(\det(A) = -1\) (improper
rotations, that is, rotations composed with a reflection). Since \(\det\) is continuous and
takes the two values \(\pm 1\) on \(O(n)\), no path inside \(O(n)\) can join one piece to the
other, so \(O(n)\) is disconnected. The connectedness proof below shows that the piece with
\(\det(A) = +1\) is connected. The other piece is its image under multiplication by a fixed
reflection, and that map is a homeomorphism of \(O(n)\), so it is connected as well. The two
pieces are therefore exactly the
connected components
of \(O(n)\), and the one containing the identity is the special orthogonal group.
Definition: Special Orthogonal Group
The special orthogonal group is
\[
\begin{align*}
SO(n) &= O(n) \cap SL(n, \mathbb{R}) \\\\
&= \{ A \in GL(n, \mathbb{R}) : A^\top A = I, \quad \det(A) = 1 \}.
\end{align*}
\]
This is the group of rotations of \(\mathbb{R}^n\). It is a closed
subgroup of \(GL(n, \mathbb{R})\) (as the intersection of two closed subgroups), and it is
compact (as a closed subset of the compact set \(O(n)\)). Its dimension is \(n(n-1)/2\).
Theorem: Euler's Rotation Theorem
Let \(A \in SO(3)\). There are an orthonormal basis
\(\{\hat{\mathbf{n}}, \mathbf{u}, \mathbf{v}\}\) of \(\mathbb{R}^3\) and an angle
\(\theta \in \mathbb{R}\) such that \(A\hat{\mathbf{n}} = \hat{\mathbf{n}}\) and
\(A\) rotates the plane spanned by \(\mathbf{u}\) and \(\mathbf{v}\) through
\(\theta\). Every element of \(SO(3)\) is therefore a rotation by some angle about
some axis through the origin.
Proof:
Step 1: the fixed axis. From \(A^\top A = I\) we get
\(A^\top - I = A^\top - A^\top A = A^\top(I - A)\), so
\[
\begin{align*}
\det(A - I) &= \det\bigl((A - I)^\top\bigr) = \det(A^\top - I) \\\\
&= \det(A^\top)\det(I - A) = \det(I - A) \\\\
&= \det(-I)\det(A - I) = -\det(A - I).
\end{align*}
\]
Here \(\det(A^\top) = \det(A) = 1\) by the
determinant of a transpose,
the middle equality uses
multiplicativity of the determinant,
and \(\det(-I) = (-1)^3 = -1\) because \(-I\) is
triangular.
Hence \(2\det(A - I) = 0\), so \(A - I\) is not invertible. By the
Invertible Matrix Theorem
the equation \((A - I)\mathbf{x} = \mathbf{0}\) has a nonzero solution
\(\mathbf{x}\), and \(\hat{\mathbf{n}} = \mathbf{x}/\|\mathbf{x}\|\) is a unit
vector with \(A\hat{\mathbf{n}} = \hat{\mathbf{n}}\).
Step 2: an adapted orthonormal basis. The set
\(\{\hat{\mathbf{n}}\}\) is linearly independent, so it
extends to a basis
of \(\mathbb{R}^3\), and the
Gram-Schmidt process
turns that basis into an orthogonal one whose first vector is again
\(\hat{\mathbf{n}}\). Scaling the other two to unit length gives an orthonormal
basis \(\{\hat{\mathbf{n}}, \mathbf{u}, \mathbf{v}\}\) of \(\mathbb{R}^3\). Let
\(Q\) be the matrix whose columns are these three vectors. Then
\(Q^\top Q = I\),
and the columns of \(Q\) are linearly independent, so \(Q\) is invertible and
\(Q^{-1} = Q^\top\).
Step 3: the matrix of \(A\) in that basis. Put
\(\tilde{A} = Q^\top A Q\). Then
\(\tilde{A}^\top\tilde{A} = Q^\top A^\top Q Q^\top A Q = Q^\top A^\top A Q = Q^\top Q = I\),
so \(\tilde{A}\) has orthonormal columns, and
\(\tilde{A}\mathbf{e}_1 = Q^\top A\hat{\mathbf{n}} = Q^\top\hat{\mathbf{n}} = \mathbf{e}_1\).
The first column of \(\tilde{A}\) is therefore \(\mathbf{e}_1\), and the other two
columns are orthogonal to it, so their first entries vanish:
\[
\tilde{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & b_{11} & b_{12} \\ 0 & b_{21} & b_{22} \end{pmatrix}.
\]
Write \(B\) for the lower right \(2 \times 2\) block. Since the first row and the
first column of \(\tilde{A}\) are \(\mathbf{e}_1\), the identity
\(\tilde{A}^\top\tilde{A} = I\) restricts to \(B^\top B = I\), and the cofactor
expansion of \(\det\tilde{A}\) along the first row gives
\(\det\tilde{A} = \det B\). On the other hand
\(\det\tilde{A} = \det(Q^\top)\det(A)\det(Q) = \det(A) = 1\), because
\(\det(Q^\top)\det(Q) = \det(Q^\top Q) = 1\). Hence \(\det B = 1\).
Step 4: the block is a plane rotation. The first column of \(B\)
is a unit vector, hence equals \((\cos\theta, \sin\theta)\) for some
\(\theta \in \mathbb{R}\). The second column is a unit vector orthogonal to the
first, so it is \(\pm(-\sin\theta, \cos\theta)\), and these two choices give
\(\det B = 1\) and \(\det B = -1\) respectively. Since \(\det B = 1\),
\[
B = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}.
\]
So \(A\) fixes \(\hat{\mathbf{n}}\) and rotates the plane spanned by
\(\mathbf{u}\) and \(\mathbf{v}\) through \(\theta\).
Proof that \(SO(n)\) is connected:
We show that \(SO(n)\) is path-connected, which
implies connectedness.
We prove this explicitly for \(n = 2\) and \(n = 3\), and then treat general
\(n\) by induction.
Case \(n = 2\). Every element of \(SO(2)\) is a rotation matrix
\(R(\theta) = \bigl(\begin{smallmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{smallmatrix}\bigr)\)
for some \(\theta \in \mathbb{R}\). The path
\(t \mapsto R(t\theta)\), \(t \in [0, 1]\), is continuous in \(SO(2)\) and connects
\(I = R(0)\) to \(R(\theta)\).
Case \(n = 3\). By
Euler's rotation theorem,
every \(A \in SO(3)\) is a rotation by some angle \(\theta\) about some unit axis
\(\hat{\mathbf{n}} \in \mathbb{R}^3\). Writing \(A = R(\hat{\mathbf{n}}, \theta)\), the
path \(t \mapsto R(\hat{\mathbf{n}}, t\theta)\) connects \(I\) to \(A\) within \(SO(3)\).
(In the pages ahead, with the sense of \(\theta\) fixed by the right-hand rule about
\(\hat{\mathbf{n}}\), this path will be identified with the one-parameter subgroup
\(t \mapsto \exp(t\theta\,\hat{\mathbf{n}}_\times)\) generated by the
infinitesimal rotation \(\hat{\mathbf{n}}_\times\).)
General \(n\) (induction on \(n\)). We extend the geometric
argument above to a complete proof by induction. The base case \(n = 2\) is
established. Suppose \(SO(n-1)\) is path-connected, and let \(A \in SO(n)\).
We construct a continuous path in \(SO(n)\) from \(I\) to \(A\) in two stages.
Stage 1: move \(\mathbf{e}_1\) to \(A\mathbf{e}_1\). If
\(A\mathbf{e}_1 = \mathbf{e}_1\), set \(R = I\) and skip to Stage 2. Otherwise choose a
2-dimensional plane \(\mathcal{P} \subset \mathbb{R}^n\) containing both \(\mathbf{e}_1\)
and \(A\mathbf{e}_1\). If these two unit vectors are linearly independent, they span such a
plane and it is unique. If instead \(A\mathbf{e}_1 = -\mathbf{e}_1\), they span only a
line, and we take for \(\mathcal{P}\) any 2-dimensional subspace containing
\(\mathbf{e}_1\), which exists because \(n \geq 2\). Let \(R \in SO(n)\) be the rotation of
\(\mathbb{R}^n\) that acts on \(\mathcal{P}\) as the rotation through the angle
\(\alpha \in (0, \pi]\) carrying \(\mathbf{e}_1\) to \(A\mathbf{e}_1\), and as the identity
on \(\mathcal{P}^\perp\). The path \(t \mapsto R(t)\), where \(R(t)\) rotates
\(\mathcal{P}\) by angle \(t\alpha\) and fixes \(\mathcal{P}^\perp\), is continuous in
\(SO(n)\) and joins \(I = R(0)\) to \(R = R(1)\). This step uses the path-connectedness of
\(SO(2)\) acting on \(\mathcal{P}\). Crucially, \(R\mathbf{e}_1 = A\mathbf{e}_1\).
Stage 2: connect \(R\) to \(A\) through matrices sending \(\mathbf{e}_1\) to \(A\mathbf{e}_1\).
Both \(R\) and \(A\) send \(\mathbf{e}_1\) to \(A\mathbf{e}_1\), so the matrix
\(A R^{-1} \in SO(n)\) fixes \(A\mathbf{e}_1\). Since \(A R^{-1}\) preserves the orthogonal
decomposition \(\mathbb{R}^n = \mathbb{R}\,A\mathbf{e}_1 \oplus (A\mathbf{e}_1)^\perp\), it
acts as the identity on the first summand and as an element of
\(SO\bigl((A\mathbf{e}_1)^\perp\bigr) \cong SO(n-1)\) on the second. By the inductive
hypothesis there is a continuous path \(S(t)\) in \(SO(n-1)\) from \(I\) to
\(A R^{-1}\big|_{(A\mathbf{e}_1)^\perp}\). Embedding each \(S(t)\) into \(SO(n)\) so that
it fixes the \(A\mathbf{e}_1\)-axis pointwise gives a continuous path in \(SO(n)\) from
\(I\) to \(A R^{-1}\). Right-multiplying this path by \(R\) yields a continuous path from
\(R\) to \(A\) inside \(SO(n)\).
Concatenating the Stage 1 path (\(I \rightsquigarrow R\)) with the Stage 2 path
(\(R \rightsquigarrow A\)) produces a continuous path in \(SO(n)\) joining \(I\) to \(A\),
which completes the induction.
The dimension formula \(\dim SO(n) = n(n-1)/2\) reflects the number of independent parameters
in a skew-symmetric matrix. Equivalently, it counts the constraints imposed by
\(A^\top A = I\). That symmetric matrix equation gives \(n(n+1)/2\) equations on \(n^2\)
entries, leaving \(n^2 - n(n+1)/2 = n(n-1)/2\) degrees of freedom. For the cases of greatest
importance, \(\dim SO(2) = 1\) (one angle of rotation) and \(\dim SO(3) = 3\) (three Euler
angles, or equivalently, a rotation axis and an angle).
Definition: Unitary Group
The unitary group is
\[
U(n) = \{ A \in GL(n, \mathbb{C}) : A^* A = I \}
\]
where \(A^* = \overline{A}^\top\) denotes the conjugate transpose. This is the group of
linear transformations preserving the standard Hermitian inner product on \(\mathbb{C}^n\).
By the same argument as for \(O(n)\), \(U(n)\) is a closed, compact subgroup of
\(GL(n, \mathbb{C})\). It is connected, and its dimension is \(n^2\) (as a real
manifold).
Definition: Special Unitary Group
The special unitary group is
\[
\begin{align*}
SU(n) &= U(n) \cap SL(n, \mathbb{C}) \\\\
&= \{ A \in GL(n, \mathbb{C}) : A^* A = I, \quad \det(A) = 1 \}.
\end{align*}
\]
It is a closed, compact, connected subgroup of \(GL(n, \mathbb{C})\) of dimension
\(n^2 - 1\). In particular, \(\dim SU(2) = 2^2 - 1 = 3\), the same dimension as \(SO(3)\).
This coincidence is a first hint of the relationship between the two groups, which the Lie
correspondence will formalize as a 2:1 covering map.
Definition: Special Euclidean Group
The special Euclidean group \(SE(3)\) is the group of rigid body motions
(rotations and translations) of \(\mathbb{R}^3\). It is realized as a matrix Lie group via
the embedding into \(GL(4, \mathbb{R})\):
\[
SE(3) = \left\{ \begin{pmatrix} R & \mathbf{t} \\ \mathbf{0}^\top & 1 \end{pmatrix}
: R \in SO(3), \quad \mathbf{t} \in \mathbb{R}^3 \right\} \subset GL(4, \mathbb{R}).
\]
The group operation corresponds to composition of rigid body motions:
\[
\begin{pmatrix} R_1 & \mathbf{t}_1 \\ \mathbf{0}^\top & 1 \end{pmatrix}
\begin{pmatrix} R_2 & \mathbf{t}_2 \\ \mathbf{0}^\top & 1 \end{pmatrix}
= \begin{pmatrix} R_1 R_2 & R_1 \mathbf{t}_2 + \mathbf{t}_1 \\ \mathbf{0}^\top & 1 \end{pmatrix}.
\]
The group \(SE(3)\) is a closed, connected subgroup of \(GL(4, \mathbb{R})\), but it is
not compact (the translation component \(\mathbf{t}\) is unbounded). Its
dimension is \(6\) (\(3\) for rotation + \(3\) for translation).
The following table summarizes the classical matrix Lie groups:
| Group |
Defining Condition |
Dimension |
Connected |
Compact |
| \(GL(n, \mathbb{R})\) |
\(\det(A) \neq 0\) |
\(n^2\) |
No (2 components) |
No |
| \(SL(n, \mathbb{R})\) |
\(\det(A) = 1\) |
\(n^2 - 1\) |
Yes |
No |
| \(O(n)\) |
\(A^\top A = I\) |
\(n(n-1)/2\) |
No (2 components) |
Yes |
| \(SO(n)\) |
\(A^\top A = I, \quad \det(A) = 1\) |
\(n(n-1)/2\) |
Yes |
Yes |
| \(U(n)\) |
\(A^* A = I\) |
\(n^2\) |
Yes |
Yes |
| \(SU(n)\) |
\(A^* A = I, \quad \det(A) = 1\) |
\(n^2 - 1\) |
Yes |
Yes |
| \(SE(3)\) |
\(\begin{pmatrix} R & \mathbf{t} \\ \mathbf{0}^\top & 1 \end{pmatrix}, \quad R \in SO(3)\) |
6 |
Yes |
No |
Cartan's Closed Subgroup Theorem
Why is closedness the only condition we need? The following theorem, due to Élie Cartan,
justifies our entire approach.
Theorem: Cartan's Closed Subgroup Theorem
Every closed subgroup of \(GL(n, \mathbb{C})\) is a smooth embedded submanifold of
\(GL(n, \mathbb{C})\), and the group operations (multiplication and inversion) are
smooth maps with respect to this manifold structure.
The proof requires the inverse function theorem on manifolds, a tool that will become available
with the future treatment of smooth manifolds. We state Cartan's theorem here as the
foundational result that justifies our definition. Calling a closed subgroup of
\(GL(n, \mathbb{C})\) a "matrix Lie group" is not merely a convention. The theorem
guarantees that it is genuinely a Lie group in the abstract sense.
Quotient Spaces
If \(H \leq G\) is a closed subgroup of a matrix Lie group \(G\), then the quotient space
\(G/H\) carries a natural smooth manifold structure (by a generalization of Cartan's theorem).
Here, \(G/H\) denotes the set of left cosets \(\{gH : g \in G\}\) equipped with the
quotient topology.
When \(H\) is a
normal subgroup,
the quotient \(G/H\) is both a smooth manifold and a group (a
factor group).
When \(H\) is merely closed, \(G/H\) is still a smooth manifold, called a
homogeneous space, but unless \(H\) is normal, multiplication of cosets is
not well defined on it.
Example: The 2-Sphere as a Homogeneous Space
The group \(SO(3)\) acts transitively on the unit sphere \(S^2 \subset \mathbb{R}^3\) (any
unit vector can be rotated to any other). The stabilizer of the north pole
\(\mathbf{e}_3 = (0, 0, 1)^\top\) consists of all rotations that fix \(\mathbf{e}_3\).
These are precisely the matrices of the form
\[
\begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}
= \begin{pmatrix} R_{2\times 2} & \mathbf{0} \\ \mathbf{0}^\top & 1 \end{pmatrix},
\]
where the upper-left \(2 \times 2\) block \(R_{2\times 2}\) ranges over \(SO(2)\).
Therefore:
\[
SO(3) / SO(2) \cong S^2.
\]
The 2-sphere is a homogeneous space. It is a smooth manifold but not a group, since there
is no natural way to "multiply" two points on a sphere.
Looking Ahead
We have established \(GL\), \(SL\), \(O\), \(SO\), \(U\), \(SU\), and \(SE(3)\) as closed
subgroups of the general linear group. Cartan's theorem assures us that each one is a smooth
manifold with smooth group operations. The summary table in the previous section makes a
recurring pattern visible. Each group is carved out of \(GL(n)\) by a set of nonlinear
algebraic equations such as \(A^\top A = I\) and \(\det(A) = 1\).
A natural question arises: is there a systematic way to move between these nonlinear
group-level constraints and simpler, linear conditions? The answer is yes, and the
tool is the matrix exponential. This power series converts matrices satisfying
linear conditions (for example, skew-symmetry \(A^\top = -A\)) into group elements satisfying
the corresponding nonlinear ones (for example, orthogonality \(e^A (e^A)^\top = I\)).
In The Matrix Exponential, we define
this exponential map, prove its fundamental properties, and use it to derive explicit formulas
for rotations, including Rodrigues' rotation formula, a standard tool for 3D
rotation in robotics and computer graphics.
The Road to Lie Algebras
The matrix exponential will reveal that each Lie group \(G\) has an associated
Lie algebra \(\mathfrak{g}\), a vector space of "infinitesimal
generators" equipped with an operation called the Lie bracket that
encodes the group's non-commutativity at the linear level. This linearization is the key
to making Lie groups computationally tractable. Instead of working with nonlinear group
elements, we work with their linear Lie algebra counterparts and exponentiate back when
needed. This theory is developed over the next three pages:
The Matrix Exponential: the bridge from linear to nonlinear.
Lie Algebras and the Lie Bracket: the tangent space at the identity and
its algebraic structure.
The Lie Correspondence: how the algebra determines the group near the
identity, the Baker-Campbell-Hausdorff formula, and the adjoint representations.