Besides just preserving length, rotations also preserve the angles between vectors. This follows from the fact that the standard dot product between two vectors u and v can be written purely in terms of length:
u
⋅
v
=
1
2
(
∥
u
+
v
∥
2
−
∥
u
∥
2
−
∥
v
∥
2
)
.
It follows that any lengthpreserving transformation in R^{3} preserves the dot product, and thus the angle between vectors. Rotations are often defined as linear transformations that preserve the inner product on R^{3}, which is equivalent to requiring them to preserve length. See classical group for a treatment of this more general approach, where SO(3) appears as a special case.
Every rotation maps an orthonormal basis of R^{3} to another orthonormal basis. Like any linear transformation of finitedimensional vector spaces, a rotation can always be represented by a matrix. Let R be a given rotation. With respect to the standard basis e_{1}, e_{2}, e_{3} of R^{3} the columns of R are given by (Re_{1}, Re_{2}, Re_{3}). Since the standard basis is orthonormal, and since R preserves angles and length, the columns of R form another orthonormal basis. This orthonormality condition can be expressed in the form
R
T
R
=
R
R
T
=
I
,
where R^{T} denotes the transpose of R and I is the 3 × 3 identity matrix. Matrices for which this property holds are called orthogonal matrices. The group of all 3 × 3 orthogonal matrices is denoted O(3), and consists of all proper and improper rotations.
In addition to preserving length, proper rotations must also preserve orientation. A matrix will preserve or reverse orientation according to whether the determinant of the matrix is positive or negative. For an orthogonal matrix R, note that det R^{T} = det R implies (det R)^{2} = 1, so that det R = ±1. The subgroup of orthogonal matrices with determinant +1 is called the special orthogonal group, denoted SO(3).
Thus every rotation can be represented uniquely by an orthogonal matrix with unit determinant. Moreover, since composition of rotations corresponds to matrix multiplication, the rotation group is isomorphic to the special orthogonal group SO(3).
Improper rotations correspond to orthogonal matrices with determinant −1, and they do not form a group because the product of two improper rotations is a proper rotation.
The rotation group is a group under function composition (or equivalently the product of linear transformations). It is a subgroup of the general linear group consisting of all invertible linear transformations of the real 3space R^{3}.
Furthermore, the rotation group is nonabelian. That is, the order in which rotations are composed makes a difference. For example, a quarter turn around the positive xaxis followed by a quarter turn around the positive yaxis is a different rotation than the one obtained by first rotating around y and then x.
The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.
Every nontrivial proper rotation in 3 dimensions fixes a unique 1dimensional linear subspace of R^{3} which is called the axis of rotation (this is Euler's rotation theorem). Each such rotation acts as an ordinary 2dimensional rotation in the plane orthogonal to this axis. Since every 2dimensional rotation can be represented by an angle φ, an arbitrary 3dimensional rotation can be specified by an axis of rotation together with an angle of rotation about this axis. (Technically, one needs to specify an orientation for the axis and whether the rotation is taken to be clockwise or counterclockwise with respect to this orientation).
For example, counterclockwise rotation about the positive zaxis by angle φ is given by
R
z
(
φ
)
=
[
cos
φ
−
sin
φ
0
sin
φ
cos
φ
0
0
0
1
]
.
Given a unit vector n in R^{3} and an angle φ, let R(φ, n) represent a counterclockwise rotation about the axis through n (with orientation determined by n). Then
R(0, n) is the identity transformation for any n
R(φ, n) = R(−φ, −n)
R(π + φ, n) = R(π − φ, −n).
Using these properties one can show that any rotation can be represented by a unique angle φ in the range 0 ≤ φ ≤ π and a unit vector n such that
n is arbitrary if φ = 0
n is unique if 0 < φ < π
n is unique up to a sign if φ = π (that is, the rotations R(π, ±n) are identical).
The Lie group SO(3) is diffeomorphic to the real projective space RP^{3}.
Consider the solid ball in R^{3} of radius π (that is, all points of R^{3} of distance π or less from the origin). Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin. The identity rotation corresponds to the point at the center of the ball. Rotation through angles between 0 and −π correspond to the point on the same axis and distance from the origin but on the opposite side of the origin. The one remaining issue is that the two rotations through π and through −π are the same. So we identify (or "glue together") antipodal points on the surface of the ball. After this identification, we arrive at a topological space homeomorphic to the rotation group.
Indeed, the ball with antipodal surface points identified is a smooth manifold, and this manifold is diffeomorphic to the rotation group. It is also diffeomorphic to the real 3dimensional projective space RP^{3}, so the latter can also serve as a topological model for the rotation group.
These identifications illustrate that SO(3) is connected but not simply connected. As to the latter, in the ball with antipodal surface points identified, consider the path running from the "north pole" straight through the interior down to the south pole. This is a closed loop, since the north pole and the south pole are identified. This loop cannot be shrunk to a point, since no matter how you deform the loop, the start and end point have to remain antipodal, or else the loop will "break open". In terms of rotations, this loop represents a continuous sequence of rotations about the zaxis starting and ending at the identity rotation (i.e. a series of rotation through an angle φ where φ runs from 0 to 2π).
Surprisingly, if you run through the path twice, i.e., run from north pole down to south pole, jump back to the north pole (using the fact that north and south poles are identified), and then again run from north pole down to south pole, so that φ runs from 0 to 4π, you get a closed loop which can be shrunk to a single point: first move the paths continuously to the ball's surface, still connecting north pole to south pole twice. The second half of the path can then be mirrored over to the antipodal side without changing the path at all. Now we have an ordinary closed loop on the surface of the ball, connecting the north pole to itself along a great circle. This circle can be shrunk to the north pole without problems. The Balinese plate trick and similar tricks demonstrate this practically.
The same argument can be performed in general, and it shows that the fundamental group of SO(3) is cyclic group of order 2. In physics applications, the nontriviality of the fundamental group allows for the existence of objects known as spinors, and is an important tool in the development of the spinstatistics theorem.
The universal cover of SO(3) is a Lie group called Spin(3). The group Spin(3) is isomorphic to the special unitary group SU(2); it is also diffeomorphic to the unit 3sphere S^{3} and can be understood as the group of versors (quaternions with absolute value 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations. The map from S^{3} onto SO(3) that identifies antipodal points of S^{3} is a surjective homomorphism of Lie groups, with kernel {±1}. Topologically, this map is a twotoone covering map. (See the plate trick.)
The general reference for this section is Gelfand, Minlos & Shapiro (1963). The points P on the sphere S = {(x, y, z) ∈ ℝ^{3}: x^{2} + y^{2} + z^{2} = 1/4} can, barring the north pole N, be put into onetoone bijection with points S(P) = P´ on the plane M defined by z = −1/2, see figure. The map S is called stereographic projection.
Let the coordinates on M be (ξ, η). The line L passing through N and P can be written
L
=
N
+
t
(
N
−
P
)
=
(
0
,
0
,
1
/
2
)
+
t
(
(
0
,
0
,
1
/
2
)
−
(
x
,
y
,
z
)
)
,
t
∈
R
.
Demanding that the zcoordinate equals −1/2, one finds t = 1/z − ^{1}⁄_{2}, hence
S
:
S
→
M
;
P
↦
P
′
;
(
x
,
y
,
z
)
↦
(
ξ
,
η
)
=
(
x
1
2
−
z
,
y
1
2
−
z
)
≡
ζ
=
ξ
+
i
η
,
where, for later convenience, the plane M is identified with the complex plane ℂ.
For the inverse, write L as
L
=
N
+
s
(
P
′
−
N
)
=
(
0
,
0
,
1
2
)
+
s
(
(
ξ
,
η
,
−
1
2
)
−
(
0
,
0
,
1
2
)
)
,
and demand x^{2} + y^{2} + z^{2} = 1/4 to find s = 1/1 + ξ^{2} + η^{2} and thus
S
−
1
:
M
→
S
;
P
′
↦
P
;
(
ξ
,
η
)
↦
(
x
,
y
,
z
)
=
(
ξ
1
+
ξ
2
+
η
2
,
η
1
+
ξ
2
+
η
2
,
−
1
+
ξ
2
+
η
2
2
+
2
ξ
2
+
2
η
2
)
.
If g ∈ SO(3) is a rotation, then it will take points on S to points on S by its standard action Π_{s}(g) on the embedding space ℝ^{3}. By composing this action with S one obtains a transformation S ∘ Π_{s}(g) ∘ S^{−1} of M, ζ = P´ ↦ P ↦ Π_{s}(g)P =gP ↦ S(gP) ≡ Π_{u}(g)ζ = ζ´. Thus Π_{u}(g) is a transformation of ℂ associated to the transformation Π_{s}(g) of ℝ^{3}.
It turns out that g ∈ SO(3) represented in this way by Π_{u}(g) can be expressed as a matrix Π_{u}(g) ∈ SU(2) (where the notation is recycled to use the same name for the matrix as for the transformation of ℂ it represents). To identify this matrix, consider first a rotation g_{φ} about the zaxis through an angle φ,
x
′
=
x
cos
φ
−
y
sin
φ
,
y
′
=
x
sin
φ
+
y
cos
φ
,
z
′
=
z
.
Hence
ζ
′
=
x
′
+
i
y
′
1
2
−
z
′
=
e
i
φ
(
x
+
i
y
)
1
2
−
z
=
e
i
φ
ζ
=
e
i
φ
2
ζ
+
0
0
ζ
+
e
−
i
φ
2
,
which, unsurprisingly, is a rotation in the complex plane. In an analogous way, if g_{θ} is a rotation about the xaxis through and angle θ, then
w
′
=
e
i
θ
w
,
w
=
y
+
i
z
1
2
−
x
,
which, after a little algebra, becomes
ζ
′
=
cos
θ
2
ζ
+
i
sin
θ
2
i
sin
θ
2
ζ
+
cos
θ
2
.
These two rotations, g_{φ}, g_{θ}, thus correspond to bilinear transforms of ℝ^{2} ≃ ℂ ≃ M, namely, they are examples of Möbius transformations.
A general Möbius transformation is given by
ζ
′
=
α
ζ
+
β
γ
ζ
+
δ
,
α
δ
−
β
γ
≠
0.
.
The rotations, g_{φ}, g_{θ} generate all of SO(3) and the composition rules of the Möbius transformations show that any composition of g_{φ}, g_{θ} translates to the corresponding composition of Möbius transformations. The Möbius transformations can be represented by matrices
(
α
β
γ
δ
)
,
α
δ
−
β
γ
=
1
,
since a common factor of α, β, γ, δ cancels.
For the same reason, the matrix is not uniquely defined since multiplication by −I has no effect on either the determinant or the Möbius transformation. The composition law of Möbius transformations follow that of the corresponding matrices. The conclusion is that each Möbius transformation corresponds to two matrices g, −g ∈ SL(2, ℂ).
Using this correspondence one may write
Π
u
(
g
φ
)
=
Π
u
[
(
cos
φ
−
sin
φ
0
sin
φ
cos
φ
0
0
0
1
)
]
=
±
(
e
i
φ
2
0
0
e
−
i
φ
2
)
,
Π
u
(
g
θ
)
=
Π
u
[
(
1
0
0
0
cos
θ
−
sin
θ
0
sin
θ
cos
θ
)
]
=
±
(
cos
θ
2
i
sin
θ
2
i
sin
θ
2
cos
θ
2
)
.
These matrices are unitary and thus Π_{u}(SO(3)) ⊂ SU(2) ⊂ SL(2, ℂ). In terms of Euler angles one finds for a general rotation
one has
For the converse, consider a general matrix
±
Π
u
(
g
α
,
β
)
=
±
(
α
β
−
β
¯
α
¯
)
∈
S
U
(
2
)
.
Make the substitutions
cos
θ
2
=

α

,
sin
θ
2
=

β

,
(
0
≤
θ
≤
π
)
,
φ
+
ψ
2
=
arg
α
,
ψ
−
φ
2
=
arg
β
.
With the substitutions, Π(g_{α, β}) assumes the form of the right hand side (RHS) of (2), which corresponds under Π_{u} to a matrix on the form of the RHS of (1) with the same φ, θ, ψ. In terms of the complex parameters α, β,
g
α
,
β
=
(
1
2
(
α
2
−
β
2
+
α
2
¯
−
β
2
¯
)
i
2
(
−
α
2
−
β
2
+
α
2
¯
+
β
2
¯
)
−
α
β
−
α
¯
β
¯
i
2
(
α
2
−
β
2
−
α
2
¯
+
β
2
¯
)
1
2
(
α
2
+
β
2
+
α
2
¯
+
β
2
¯
)
−
i
(
+
α
β
−
α
¯
β
¯
)
α
β
¯
+
α
¯
β
i
(
−
α
β
¯
+
α
¯
β
)
α
α
¯
−
β
β
¯
)
.
To verify this, substitute for α. β the elements of the matrix on the RHS of (2). After some manipulation, the matrix assumes the form of the RHS of (1).
It is clear from the explicit form in terms of Euler angles that the map p:SU(2) → SO(3);Π(±g_{αβ}) ↦ g_{αβ} just described is a smooth, 2:1 and onto group homomorphism. It is hence an explicit description of the universal covering map of SO(3) from the universal covering group SU(2).
SU(2) is isomorphic to the quaternions of unit norm via a map given by
q
=
a
1
+
b
i
+
c
j
+
d
k
=
α
+
j
β
↔
[
α
−
β
¯
β
α
¯
]
=
U
,
q
∈
H
,
a
,
b
,
c
,
d
∈
R
,
α
,
β
∈
C
,
U
∈
S
U
(
2
)
.
This means that there is a 2:1 homomorphism from quaternions of unit norm to SO(3). Concretely, a unit quaternion, q, with
q
=
w
+
i
x
+
j
y
+
k
z
,
1
=
w
2
+
x
2
+
y
2
+
z
2
,
is mapped to the rotation matrix
Q
=
[
1
−
2
y
2
−
2
z
2
2
x
y
−
2
z
w
2
x
z
+
2
y
w
2
x
y
+
2
z
w
1
−
2
x
2
−
2
z
2
2
y
z
−
2
x
w
2
x
z
−
2
y
w
2
y
z
+
2
x
w
1
−
2
x
2
−
2
y
2
]
.
This is a rotation around the vector (x,y,z) by an angle 2θ, where cos θ = w and sin θ = (x,y,z). The proper sign for sin θ is implied, once the signs of the axis components are fixed. The 2:1nature is apparent since both q and −q map to the same Q.
Associated with every Lie group is its Lie algebra, a linear space of the same dimension as the Lie group, closed under a bilinear alternating product called the Lie bracket. The Lie algebra of SO(3) is denoted by so(3) and consists of all skewsymmetric 3 × 3 matrices. This may be seen by differentiating the orthogonality condition, A^{T}A = I, A ∈ SO(3). The Lie bracket of two elements of so(3) is, as for the Lie algebra of every matrix group, given by the matrix commutator, [A_{1}, A_{2}] = A_{1}A_{2} − A_{2}A_{1}, which is again a skewsymmetric matrix. The Lie algebra bracket captures the essence of the Lie group product in a sense made precise by the Baker–Campbell–Hausdorff formula.
The elements of so(3) are the "infinitesimal generators" of rotations, i.e. they are the elements of the tangent space of the manifold SO(3) at the identity element. If R(φ, n) denotes a counterclockwise rotation with angle φ about the axis specified by the unit vector n, then
d
d
ϕ

ϕ
=
0
R
(
ϕ
,
n
)
x
=
n
×
x
for every vector x in R^{3}.
This can be used to show that the Lie algebra so(3) (with commutator) is isomorphic to the Lie algebra R^{3} (with cross product). Under this isomorphism, an Euler vector
ω
∈
R
3
corresponds to the linear map
ω
~
defined by
ω
~
(
x
)
=
ω
×
x
.
In more detail, a most often suitable basis for so(3) as a 3dimensional vector space is
L
x
=
[
0
0
0
0
0
−
1
0
1
0
]
,
L
y
=
[
0
0
1
0
0
0
−
1
0
0
]
,
L
z
=
[
0
−
1
0
1
0
0
0
0
0
]
.
The commutation relations of these basis elements are,
[
L
x
,
L
y
]
=
L
z
,
[
L
z
,
L
x
]
=
L
y
,
[
L
y
,
L
z
]
=
L
x
which agree with the relations of the three standard unit vectors of R^{3} under the cross product.
As announced above, one can identify any matrix in this Lie algebra with an Euler vector in ℝ^{3},
ω
=
(
x
,
y
,
z
)
∈
R
3
,
ω
~
=
ω
⋅
L
=
x
L
x
+
y
L
y
+
z
L
z
=
[
0
−
z
y
z
0
−
x
−
y
x
0
]
∈
s
o
(
3
)
.
This identification is sometimes called the hatmap. Under this identification, the so(3) bracket corresponds in ℝ^{3} to the cross product,
[
u
~
,
v
~
]
=
u
×
v
~
.
The matrix identified with a vector u has the property that
u
~
v
=
u
×
v
,
where ordinary matrix multiplication is implied on the left hand side. This implies that u is in the null space of the skewsymmetric matrix with which it is identified, because u × u = 0.
In Lie algebra representation, the group SO(3) is compact and simple of rank 1, and so it has a single independent Casimir element, a quadratic invariant function of the three generators which commutes with all of them. The Killing form for the rotation group is just the Kronecker delta, and so this Casimir invariant is simply the sum of the squares of the generators,
J
x
,
J
y
,
J
z
, of the algebra
[
J
x
,
J
y
]
=
J
z
,
[
J
z
,
J
x
]
=
J
y
,
[
J
y
,
J
z
]
=
J
x
.
That is, the Casimir invariant is given by
J
2
≡
J
⋅
J
=
J
x
2
+
J
y
2
+
J
z
2
∝
I
.
For unitary irreducible representations D^{j}, the eigenvalues of this invariant are real and discrete, and characterize each representation, which is finite dimensional, of dimensionality 2j+1. That is, the eigenvalues of this Casimir operator are
J
2
=
−
j
(
j
+
1
)
I
2
j
+
1
,
where j is integer or halfinteger, and referred to as the spin or angular momentum.
So, above, the 3×3 generators L displayed act on the triplet (spin 1) representation, while the 2×2 ones (t) act on the doublet (spin½) representation. By taking Kronecker products of D^{1/2} with itself repeatedly, one may construct all higher irreducible representations D^{j}. That is, the resulting generators for higher spin systems in three spatial dimensions, for arbitrarily large j, can be calculated using these spin operators and ladder operators.
For every unitary irreducible representations D^{j} there is an equivalent one, D^{−j−1}. All infinitedimensional irreducible representations must be nonunitary, since the group is compact.
In quantum mechanics, the Casimir invariant is the "angularmomentumsquared" operator; integer values of spin j characterize bosonic representations, while halfinteger values fermionic representations, respectively. The antihermitian matrices used above are utilized as spin operators, after they are multiplied by i, so they are now hermitian (like the Pauli matrices). Thus, in this language,
[
J
x
,
J
y
]
=
i
J
z
,
[
J
z
,
J
x
]
=
i
J
y
,
[
J
y
,
J
z
]
=
i
J
x
.
and hence
J
2
=
j
(
j
+
1
)
I
2
j
+
1
.
Explicit expressions for these D^{j} are,
(
J
z
(
j
)
)
b
a
=
(
j
+
1
−
a
)
δ
b
,
a
(
J
x
(
j
)
)
b
a
=
1
2
(
δ
b
,
a
+
1
+
δ
b
+
1
,
a
)
(
j
+
1
)
(
a
+
b
−
1
)
−
a
b
(
J
y
(
j
)
)
b
a
=
1
2
i
(
δ
b
,
a
+
1
−
δ
b
+
1
,
a
)
(
j
+
1
)
(
a
+
b
−
1
)
−
a
b
1
≤
a
,
b
≤
2
j
+
1
,
for arbitrary j.
For example, the resulting spin matrices for spin 1, spin 3/2, and 5/2 are:
For
j
=
1
J
x
=
1
2
(
0
1
0
1
0
1
0
1
0
)
J
y
=
1
2
(
0
−
i
0
i
0
−
i
0
i
0
)
J
z
=
(
1
0
0
0
0
0
0
0
−
1
)
(Note, however, how these are in an equivalent, but different basis than the above i Ls.)
For
j
=
3
2
:
J
x
=
1
2
(
0
3
0
0
3
0
2
0
0
2
0
3
0
0
3
0
)
J
y
=
1
2
(
0
−
i
3
0
0
i
3
0
−
2
i
0
0
2
i
0
−
i
3
0
0
i
3
0
)
J
z
=
1
2
(
3
0
0
0
0
1
0
0
0
0
−
1
0
0
0
0
−
3
)
.
For
j
=
5
2
:
J
x
=
1
2
(
0
5
0
0
0
0
5
0
2
2
0
0
0
0
2
2
0
3
0
0
0
0
3
0
2
2
0
0
0
0
2
2
0
5
0
0
0
0
5
0
)
J
y
=
1
2
(
0
−
i
5
0
0
0
0
i
5
0
−
2
i
2
0
0
0
0
2
i
2
0
−
3
i
0
0
0
0
3
i
0
−
2
i
2
0
0
0
0
2
i
2
0
−
i
5
0
0
0
0
i
5
0
)
J
z
=
1
2
(
5
0
0
0
0
0
0
3
0
0
0
0
0
0
1
0
0
0
0
0
0
−
1
0
0
0
0
0
0
−
3
0
0
0
0
0
0
−
5
)
.
and so on.
The Lie algebras so(3) and su(2) are isomorphic. One basis for su(2) is given by
t
1
=
1
2
[
0
−
i
−
i
0
]
,
t
2
=
1
2
[
0
−
1
1
0
]
,
t
3
=
1
2
[
−
i
0
0
i
]
.
These are related to the Pauli matrices by t_{i} ↔ 1/2iσ_{i}. The Pauli matrices abide the physicist convention for Lie algebras. In that convention, Lie algebra elements are multiplied by i, the exponential map (below) is defined with an extra factor of i in the exponent and the structure constants remain the same, but the definition of them acquires a factor of i. Likewise, commutation relations acquire a factor of i. The commutation relations for the t_{i} are
[
t
i
,
t
j
]
=
ϵ
i
j
k
t
k
,
where ε_{ijk} is the totally antisymmetric symbol with ε_{123} = 1. The isomorphism between so(3) and su(2) can be set up in several ways. For later convenience, so(3) and su(2) are identified by mapping
L
x
↔
t
1
,
L
y
↔
t
2
,
L
z
↔
t
3
,
and extending by linearity.
The exponential map for SO(3), is, since SO(3) is a matrix Lie group, defined using the standard matrix exponential series,
exp
:
s
o
(
3
)
→
S
O
(
3
)
;
A
↦
e
A
=
∑
k
=
0
∞
1
k
!
A
k
=
I
+
A
+
1
2
A
2
+
⋯
.
For any skewsymmetric matrix A ∈ so(3), e^{A} is always in SO(3). The level of difficulty of proof depends on how a matrix group Lie algebra is defined. Hall (2003) defines the Lie algebra as the set of matrices A ∈ M_{n}(ℝ) e^{tA} ∈ SO(3) ∀t, in which case it is trivial. Rossmann (2002) uses for a definition derivatives of smooth curve segments in SO(3) through the identity taken at the identity, in which case it is harder.
For a fixed A ≠ 0, e^{tA}, −∞ < t < ∞ is a oneparameter subgroup along a geodesic in SO(3). That this gives a oneparameter subgroup follows directly from properties of the exponential map.
The exponential map provides a diffeomorphism between a neighborhood of the origin in the so(3) and a neighborhood of the identity in the SO(3). For a proof, see Closed subgroup theorem.
The exponential map is surjective. This follows from the fact that every R ∈ SO(3), since every rotation leaves an axis fixed (Euler's rotation theorem), and is conjugate to a block diagonal matrix of the form
D
=
(
cos
θ
−
sin
θ
0
sin
θ
cos
θ
0
0
0
1
)
=
e
θ
L
z
,
such that A = BDB^{−1}, and that
B
e
θ
L
z
B
−
1
=
e
B
θ
L
z
B
−
1
,
together with the fact that so(3) is closed under the adjoint action of SO(3), meaning that BθL_{z}B^{−1} ∈ so(3).
Thus, e.g., it is easy to check the popular identity
e
−
π
L
x
/
2
e
θ
L
z
e
π
L
x
/
2
=
e
θ
L
y
.
As shown above, every element A ∈ so(3) is associated with a vector ω = θ u, where u = (x,y,z) is a unit magnitude vector. Since u is in the null space of A, if one now rotates to a new basis, through some other orthogonal matrix O, with u as the z axis, the final column and row of the rotation matrix in the new basis will be zero.
Thus, we know in advance from the formula for the exponential that exp(OAO^{T}) must leave u fixed. It is mathematically impossible to supply a straightforward formula for such a basis as a function of u, because its existence would violate the hairy ball theorem; but direct exponentiation is possible, and yields
exp
(
ω
~
)
=
exp
(
[
0
−
z
θ
y
θ
z
θ
0
−
x
θ
−
y
θ
x
θ
0
]
)
=
I
+
2
c
s
u
⋅
L
+
2
s
2
(
u
⋅
L
)
2
=
[
2
(
x
2
−
1
)
s
2
+
1
2
x
y
s
2
−
2
z
c
s
2
x
z
s
2
+
2
y
c
s
2
x
y
s
2
+
2
z
c
s
2
(
y
2
−
1
)
s
2
+
1
2
y
z
s
2
−
2
x
c
s
2
x
z
s
2
−
2
y
c
s
2
y
z
s
2
+
2
x
c
s
2
(
z
2
−
1
)
s
2
+
1
]
,
where c = cos ^{θ}⁄_{2}, s = sin ^{θ}⁄_{2}. This is recognized as a matrix for a rotation around axis u by the angle θ: cf. Rodrigues' rotation formula.
Given R ∈ SO(3), let
A
=
R
−
R
T
2
denote the antisymmetric part.
Then, the logarithm of A is given by
log
R
=
sin
−
1


A




A


A
.
This is manifest by inspection of the mixed symmetry form of Rodrigues' formula,
e
X
=
I
+
sin
θ
θ
X
+
2
sin
2
θ
2
θ
2
X
2
,
θ
=


X


,
where the first and last term on the righthand side are symmetric.
Suppose X and Y in the Lie algebra are given. Their exponentials, exp(X) and exp(Y), are rotation matrices, which can be multiplied. Since the exponential map is a surjection, for some Z in the Lie algebra, exp(Z) = exp(X) exp(Y), and one may tentatively write
Z
=
C
(
X
,
Y
)
,
for C some expression in X and Y. When exp(X) and exp(Y) commute, then Z = X + Y, mimicking the behavior of complex exponentiation.
The general case is given by the more elaborate BCH formula, a series expansion of nested Lie brackets. For matrices, the Lie bracket is the same operation as the commutator, which monitors lack of commutativity in multiplication. This general expansion unfolds as follows,
Z
=
C
(
X
,
Y
)
=
X
+
Y
+
1
2
[
X
,
Y
]
+
1
12
[
X
,
[
X
,
Y
]
]
−
1
12
[
Y
,
[
X
,
Y
]
]
+
⋯
.
The infinite expansion in the BCH formula for SO(3) reduces to a compact form,
Z
=
α
X
+
β
Y
+
γ
[
X
,
Y
]
,
for suitable trigonometric function coefficients (α, β, γ).
It is worthwhile to write this composite rotation generator as
α
X
+
β
Y
+
γ
[
X
,
Y
]
=
s
o
(
3
)
X
+
Y
+
1
2
[
X
,
Y
]
+
1
12
[
X
,
[
X
,
Y
]
]
−
1
12
[
Y
,
[
X
,
Y
]
]
+
⋯
,
to emphasize that this is a Lie algebra identity.
The above identity holds for all faithful representations of so(3). The kernel of a Lie algebra homomorphism is an ideal, but so(3), being simple, has no nontrivial ideals and all nontrivial representations are hence faithful. It holds in particular in the doublet or spinor representation. The same explicit formula thus follows in a simpler way through Pauli matrices, cf. the 2×2 derivation for SU(2).
For the general n × n case, one might use Ref.
The matrices in the Lie algebra are not themselves rotations; the skewsymmetric matrices are derivatives. An actual "differential rotation", or infinitesimal rotation matrix has the form
I
+
A
d
θ
,
where dθ is vanishingly small and A ∈ so(3).
These matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals . To understand what this means, one considers
d
A
x
=
[
1
0
0
0
1
−
d
θ
0
d
θ
1
]
.
First, test the orthogonality condition, Q^{T}Q = I. The product is
d
A
x
T
d
A
x
=
[
1
0
0
0
1
+
d
θ
2
0
0
0
1
+
d
θ
2
]
,
differing from an identity matrix by second order infinitesimals, discarded here. So, to first order, an infinitesimal rotation matrix is an orthogonal matrix.
Next, examine the square of the matrix,
d
A
x
2
=
[
1
0
0
0
1
−
d
θ
2
−
2
d
θ
0
2
d
θ
1
−
d
θ
2
]
.
Again discarding second order effects, note that the angle simply doubles. This hints at the most essential difference in behavior, which we can exhibit with the assistance of a second infinitesimal rotation,
d
A
y
=
[
1
0
d
ϕ
0
1
0
−
d
ϕ
0
1
]
.
Compare the products dA_{x}dA_{y} to dA_{y}dA_{x},
d
A
x
d
A
y
=
[
1
0
d
ϕ
d
θ
d
ϕ
1
−
d
θ
−
d
ϕ
d
θ
1
]
d
A
y
d
A
x
=
[
1
d
θ
d
ϕ
d
ϕ
0
1
−
d
θ
−
d
ϕ
d
θ
1
]
.
Since dθ dφ is second order, we discard it: thus, to first order, multiplication of infinitesimal rotation matrices is commutative. In fact,
d
A
x
d
A
y
=
d
A
y
d
A
x
,
again to first order. In other words, the order in which infinitesimal rotations are applied is irrelevant.
This useful fact makes, for example, derivation of rigid body rotation relatively simple. But one must always be careful to distinguish (the first order treatment of) these infinitesimal rotation matrices from both finite rotation matrices and from Lie algebra elements. When contrasting the behavior of finite rotation matrices in the BCH formula above with that of infinitesimal rotation matrices, where all the commutator terms will be second order infinitesimals one finds a bona fide vector space. Technically, this dismissal of any second order terms amounts to Group contraction.
We have seen that there are a variety of ways to represent rotations:
as orthogonal matrices with determinant 1,
by axis and rotation angle
in quaternion algebra with versors and the map 3sphere S^{3} → SO(3) (see quaternions and spatial rotations)
in geometric algebra as a rotor
as a sequence of three rotations about three fixed axes; see Euler angles.
The subgroup SO(3) of threedimensional Euclidean rotations has an infinitedimensional representation on the Hilbert space L^{2}(S^{2}) = span{Y^{ℓ}_{m}, ℓ ∈ N^{+}, −ℓ ≤ m ≤ ℓ }, where the Y^{ℓ}_{m} are spherical harmonics. Its elements are square integrable complexvalued functions on the sphere. The inner product on this space is given by
If f is an arbitrary square integrable function defined on the unit sphere S^{2}, then it can be expressed as
where the expansion coefficients are given by
The Lorentz group action restricts to that of SO(3) and is expressed as
This action is unitary, meaning that
The D^{(ℓ)} can be obtained from the D^{(m, n)} of above using Clebsch–Gordan decomposition, but they are more easily directly expressed as an exponential of an odddimensional su(2)representation (the 3dimensional one is exactly so(3)). In this case the space L^{2}(S^{2}) decomposes neatly into an infinite direct sum of irreducible odd finitedimensional representations V_{2i + 1}, i = 0, 1, … according to
This is characteristic of infinitedimensional unitary representations of SO(3). If Π is an infinitedimensional unitary representation on a separable Hilbert space, then it decomposes as a direct sum of finitedimensional unitary representations. Such a representation is thus never irreducible. All irreducible finitedimensional representations (Π, V) can be made unitary by an appropriate choice of inner product,
⟨
f
,
g
⟩
U
≡
∫
S
O
(
3
)
⟨
Π
(
R
)
f
,
Π
(
R
)
g
⟩
d
g
=
1
8
π
2
∫
0
2
π
∫
0
π
∫
0
2
π
⟨
Π
(
R
)
f
,
Π
(
R
)
g
⟩
sin
θ
d
φ
d
θ
d
ψ
,
f
,
g
∈
V
,
where the integral is the unique invariant integral over SO(3) normalized to 1, here expressed using the Euler angles parametrization. The inner product inside the integral is any inner product on V.
The rotation group generalizes quite naturally to ndimensional Euclidean space, R^{n} with its standard Euclidean structure. The group of all proper and improper rotations in n dimensions is called the orthogonal group O(n), and the subgroup of proper rotations is called the special orthogonal group SO(n), which is a Lie group of dimension n(n − 1)/2.
In special relativity, one works in a 4dimensional vector space, known as Minkowski space rather than 3dimensional Euclidean space. Unlike Euclidean space, Minkowski space has an inner product with an indefinite signature. However, one can still define generalized rotations which preserve this inner product. Such generalized rotations are known as Lorentz transformations and the group of all such transformations is called the Lorentz group.
The rotation group SO(3) can be described as a subgroup of E^{+}(3), the Euclidean group of direct isometries of Euclidean R^{3}. This larger group is the group of all motions of a rigid body: each of these is a combination of a rotation about an arbitrary axis and a translation along the axis, or put differently, a combination of an element of SO(3) and an arbitrary translation.
In general, the rotation group of an object is the symmetry group within the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.