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 — and 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:
\(SO(3)\) 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 — but we did not 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 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 \(GL(n, \mathbb{C})\). A deep theorem — 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 in our curriculum: 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 Cramer's rule as a rational function of the entries (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 \(A_1, A_2, A_3, \dots \in G\) 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 this: 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 \(A \in SL(n, \mathbb{R})\) — the group is closed.
In contrast, a sequence like \(A_k = \frac{1}{k}I\) has \(\det(A_k) = k^{-n} \to 0\),
so the 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 remarkably mild — it is the only topological condition
we impose. Yet it has profound consequences, as we will see in Cartan's theorem below.
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 \(GL(n, \mathbb{C})\).
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-preserving linear transformations.
Proof that \(SL(n, \mathbb{R})\) is a matrix Lie group:
The determinant map \(\det : GL(n, \mathbb{R}) \to \mathbb{R}^*\) is a continuous
group homomorphism
(with \(\mathbb{R}^*\) under multiplication). Its
kernel
is \(SL(n, \mathbb{R}) = \det^{-1}(\{1\})\). Since \(\{1\}\) is closed in
\(\mathbb{R}^*\) 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
\(U\) satisfying \(U^\top U = I\) — the norm-preserving transformations studied earlier in the curriculum.
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 \(A \in O(n)\) shows that \(O(n)\) has
two connected components: the matrices with \(\det(A) = +1\) (proper
rotations) and those with \(\det(A) = -1\) (improper rotations, i.e., rotations composed
with a reflection). The component containing the identity is the special orthogonal group.
Definition: Special Orthogonal Group
The special orthogonal group is
\[
SO(n) = O(n) \cap SL(n, \mathbb{R}) = \{ A \in GL(n, \mathbb{R}) : A^\top A = I,\; \det(A) = 1 \}.
\]
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
\(n(n-1)/2\).
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 comment on the general case.
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
\(SO(3)\). (We will see in the next page that this path is precisely 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
the unit vectors \(\mathbf{e}_1\) and \(A\mathbf{e}_1\) span a 2-dimensional
plane \(\mathcal{P} \subset \mathbb{R}^n\). Let \(R \in SO(n)\) be the rotation
of \(\mathbb{R}^n\) that acts as the unique angle-\(\alpha\) rotation of
\(\mathcal{P}\) sending \(\mathbf{e}_1\) to \(A\mathbf{e}_1\) (where \(\alpha\)
is the angle between them) 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
connects \(I = R(0)\) to \(R = R(1)\); this 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 \(A\) inside the stabilizer of
\(A\mathbf{e}_1\). 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 \(SO(n)\) by acting as the identity on the \(A\mathbf{e}_1\)-axis 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 (\(R \rightsquigarrow A\)) gives a continuous path in \(SO(n)\) from
\(I\) to \(A\), completing the induction.
The dimension formula \(\dim SO(n) = n(n-1)/2\) reflects the number of independent
parameters in a skew-symmetric matrix (equivalently, the number of independent constraints
in \(A^\top A = I\): the 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
\[
SU(n) = U(n) \cap SL(n, \mathbb{C}) = \{ A \in GL(n, \mathbb{C}) : A^* A = I,\; \det(A) = 1 \}.
\]
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
\(SO(3)\) — a first hint of the deep relationship between these 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 \(GL(4, \mathbb{R})\):
\[
SE(3) = \left\{ \begin{pmatrix} R & \mathbf{t} \\ \mathbf{0}^\top & 1 \end{pmatrix}
: R \in SO(3),\; \mathbf{t} \in \mathbb{R}^3 \right\} \subset GL(4, \mathbb{R}).
\]
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 |
| \(SO(n)\) |
\(A^\top A = I,\; \det(A) = 1\) |
\(n(n-1)/2\) |
Yes |
Yes |
| \(U(n)\) |
\(A^* A = I\) |
\(n^2\) |
Yes |
Yes |
| \(SU(n)\) |
\(A^* A = I,\; \det(A) = 1\) |
\(n^2 - 1\) |
Yes |
Yes |
| \(SE(3)\) |
\(\begin{pmatrix} R & \mathbf{t} \\ \mathbf{0}^\top & 1 \end{pmatrix},\; R \in SO(3)\) |
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 (not necessarily normal), \(G/H\) is still a smooth manifold —
called a homogeneous space — but it does not carry a group structure.
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 the \(z\)-axis.
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
(there is no natural way to "multiply" two points on a sphere).
Looking Ahead
We have established the classical matrix Lie groups — \(GL\), \(SL\), \(O\), \(SO\),
\(U\), \(SU\), and \(SE(3)\) — as closed subgroups of the general linear group, and
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
(\(A^\top A = I\), \(\det(A) = 1\), etc.).
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 — a power series that converts
matrices satisfying linear conditions (e.g., skew-symmetry \(A^\top = -A\)) into group
elements satisfying the corresponding nonlinear ones (e.g., orthogonality
\(e^A (e^A)^\top = I\)). In the
next page, we define this
exponential map, prove its fundamental properties, and use it to derive explicit formulas
for rotations — including Rodrigues' rotation formula, the computational
backbone of 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. The full development of this theory spans the next
three pages:
The Matrix Exponential (next) — 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,
the Baker-Campbell-Hausdorff formula, and the adjoint representations.