Rotation matrix
no illusion; not just one, but many, copies of n-dimensional rotations are found within (n+1)-dimensional rotations, as subgroups. Each embedding leaves one direction fixed, which in the case of 33 matrices is the rotation axis. For example, we have
fixing the x axis, the y axis, and the z axis, respectively. The rotation axis need not be a coordinate axis; if u = (x,y,z) is a unit vector in the desired direction, then
where c = cos , s = sin , is a rotation by angle leaving axis u fixed.
A direction in (n+1)-dimensional space will be a unit magnitude vector, which we may consider a point on a generalized sphere, Sn. Thus it is natural to describe the rotation group SO(n+1) as combining SO(n) and Sn. A suitable formalism is the fiber bundle,
where for every direction in the “base space”, Sn, the “fiber” over it in the “total space”, SO(n+1), is a copy of the “fiber space”, SO(n), namely the rotations that keep that direction fixed.
Thus we can build an nn rotation matrix by starting with a 22 matrix, aiming its fixed axis on S2 (the ordinary sphere in three-dimensional space), aiming the resulting rotation on S3, and so on up through Sn1. A point on Sn can be selected using n numbers, so we again have n(n1)/2 numbers to describe any nn rotation matrix.
In fact, we can view the sequential angle decomposition, discussed previously, as reversing this process. The composition of n1 Givens rotations brings the first column (and row) to (1,0,,0), so that the remainder of the matrix is a rotation matrix of dimension one less, embedded so as to leave (1,0,,0) fixed.
Skew parameters via Cayley’s formula
When an nn rotation matrix, Q, does not include 1 as an eigenvalue, so that none of the planar rotations of which it is composed are 180 rotations, then Q+I is an invertible matrix. Most rotation matrices fit this description, and for them we can show that (Q)(Q+I)1 is a skew-symmetric matrix, A. Thus AT = ; and since the diagonal is necessarily zero, and since the upper triangle determines the lower one, A contains n(n1)/2 independent numbers. Conveniently, I is invertible whenever A is skew-symmetric; thus we can recover the original matrix using the Cayley transform,
which maps any skew-symmetric matrix A to a rotation matrix. In fact, aside from the noted exceptions, we can produce any rotation matrix in this way. Although in practical applications we can hardly afford to ignore 180 rotations, the Cayley transform is still a potentially useful tool, giving a parameterization of most rotation matrices without trigonometric functions.
In three dimensions, for example, we have (Cayley 1846)
If we condense the skew entries into a vector, (x,y,z), then we produce a 90 rotation around the x axis for (1,0,0), around the y axis for (0,1,0), and around the z axis for (0,0,1). The 180 rotations are just out of reach; for, in the limit as x goes to infinity, (x,0,0) does approach a 180 rotation around the x axis, and similarly for other directions.
Lie theory
Lie group
We have established that nn rotation matrices form a group, the special orthogonal group, SO(n). This algebraic structure is coupled with a topological structure, in that the operations of multiplication and taking the inverse (which here is merely transposition) are continuous functions of the matrix entries. Thus SO(n) is a classic example of a topological group. (In purely topological terms, it is a compact manifold.) Furthermore, the operations are not only continuous, but smooth, so SO(n) is a differentiable manifold and a Lie group (Baker (2003); Fulton & Harris (1991)).
Most properties of rotation matrices depend very little on the dimension, n; yet in Lie group theory we see systematic differences between even dimensions and odd dimensions. As well, there are some irregularities below n = 5; for example, SO(4) is, anomalously, not a simple Lie group, but instead isomorphic to the product of S3 and SO(3).
Lie algebra
Associated with every Lie group is a Lie algebra, a linear space equipped with a bilinear alternating product called a bracket. The algebra for SO(n) is denoted by
and consists of all skew-symmetric nn matrices (as implied by differentiating the orthogonality condition, I = QTQ). The bracket, [A1,A2], of two skew-symmetric matrices is defined to be A1A22A1, which is again a skew-symmetric matrix. This Lie algebra bracket captures the essence of the Lie group product via infinitesimals.
For 22 rotation matrices, the Lie algebra is a one-dimensional vector space, multiples of
Here the bracket always vanishes, which tells us that, in two dimensions, rotations commute. Not so in any higher dimension. For 33 rotation matrices, we have a three-dimensional vector space with the convenient basis
The Lie brackets of these generators are as follows
We can conveniently identify any matrix in this Lie