Rotation matrix
algebra with a vector in R3,
Under this identification, the so(3) bracket has a memorable description; it is the vector cross product,
The matrix identified with a vector v is also memorable, because
Notice this implies that v is in the null space of the skew-symmetric matrix with which it is identified, because vv is always the zero vector.
Exponential map
Connecting the Lie algebra to the Lie group is the exponential map, which we define using the familiar power series for ex (Wedderburn 1934, 8.02),
For any skew-symmetric A, exp(A) is always a rotation matrix.
An important practical example is the 33 case, where we have seen we can identify every skew-symmetric matrix with a vector = u, where u = (x,y,z) is a unit magnitude vector. Recall that u is in the null space of the matrix associated with , so that if we use a basis with u as the z axis the final column and row will be zero. Thus we know in advance that the exponential matrix must leave u fixed. It is mathematically impossible to supply a straightforward formula for such a basis as a function of u (its existence would violate the hairy ball theorem), but direct exponentiation is possible, and yields
where c = cos 2, s = sin 2. We recognize this as our matrix for a rotation around axis u by angle . We also note that this mapping of skew-symmetric matrices is quite different from the Cayley transform discussed earlier.
In any dimension, if we choose some nonzero A and consider all its scalar multiples, exponentiation yields rotation matrices along a geodesic of the group manifold, forming a one-parameter subgroup of the Lie group. More broadly, the exponential map provides a homeomorphism between a neighborhood of the origin in the Lie algebra and a neighborhood of the identity in the Lie group. In fact, we can produce any rotation matrix as the exponential of some skew-symmetric matrix, so for these groups the exponential map is a surjection.
Bakerampbellausdorff formula
Suppose we are given A and B in the Lie algebra. Their exponentials, exp(A) and exp(B), are rotation matrices, which we can multiply. Since the exponential map is a surjection, we know that for some C in the Lie algebra, exp(A)exp(B) = exp(C), and we write
When exp(A) and exp(B) commute (which always happens for 22 matrices, but not higher), then C = A+B, mimicking the behavior of complex exponentiation. The general case is given by the BCH formula, a series expanded in terms of the bracket (Hall 2004, Ch. 3; Varadarajan 1984, 2.15). For matrices, the bracket is the same operation as the commutator, which detects lack of commutativity in multiplication. The general formula begins as follows.
Representation of a rotation matrix as a sequential angle decomposition, as in Euler angles, may tempt us to treat rotations as a vector space, but the higher order terms in the BCH formula reveal that to be a mistake.
We again take special interest in the 33 case, where [A,B] equals the cross product, AB. If A and B are linearly independent, then A, B, and AB can be used as a basis; if not, then A and B commute. And conveniently, in this dimension the summation in the BCH formula has a closed form (Eng 2001) as A+B+(AB).
Spin group
The Lie group of nn rotation matrices, SO(n), is a compact and path-connected manifold, and thus locally compact and connected. However, it is not simply connected, so Lie theory tells us it is a kind of “shadow” (a homomorphic image) of a universal covering group. Often the covering group, which in this case is the spin group denoted by Spin(n), is simpler and more natural to work with (Baker 2003, Ch. 5; Fulton & Harris 1991, pp. 299315).
In the case of planar rotations, SO(2) is topologically a circle, S1. Its universal covering group, Spin(2), is isomorphic to the real line, R, under addition. In other words, whenever we use angles of arbitrary magnitude, which we often do, we are essentially taking advantage of the convenience of the “mother space”. Every 22 rotation matrix is produced by a countable infinity of angles, separated by integer multiples of 2. Correspondingly, the fundamental group of SO(2) is isomorphic to the integers, Z.
In the case of spatial rotations, SO(3) is topologically equivalent to three-dimensional real projective space, RP3. Its universal covering group, Spin(3), is isomorphic to the 3-sphere, S3. Every 33 rotation matrix is produced by two opposite points on the sphere. Correspondingly, the fundamental group of SO(2) is isomorphic to the two-element group, Z2. We can also describe Spin(3) as isomorphic to quaternions of unit norm under multiplication, or to certain 44 real matrices, or to 22 complex special unitary matrices.
Concretely, a unit quaternion, q, with
produces the rotation matrix
This is our third version of this matrix, here as a rotation around non-unit axis vector (x,y,z) by