covering map. By properties of covering maps, the inverse can be chosen ono-to-one as a local section, but not globally. Infinitesimal rotations The matrices in the Lie algebra are not themselves rotations; the skew-symmetric matrices are derivatives, proportional differences of rotations. An actual "differential rotation", or infinitesimal rotation matrix has the form where is vanishingly small and , for instance with , The computation rules are as usual except that infinitesimals of second order are routinely dropped. With these rules, these matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals. It turns out that the order in which infinitesimal rotations are applied is irrelevant. To see this exemplified, consult infinitesimal rotations SO(3). Conversions We have seen the existence of several decompositions that apply in any dimension, namely independent planes, sequential angles, and nested dimensions. In all these cases we can either decompose a matrix or construct one. We have also given special attention to rotation matrices, and these warrant further attention, in both directions . Quaternion Given the unit quaternion , the equivalent pre-multiplied (to be used with column vectors) rotation matrix is Now every quaternion component appears multiplied by two in a term of degree two, and if all such terms are zero what is left is an identity matrix. This leads to an efficient, robust conversion from any quaternion – whether unit or non-unit – to a rotation matrix. Given: we can calculate Freed from the demand for a unit quaternion, we find that nonzero quaternions act as homogeneous coordinates for rotation matrices. The Cayley transform, discussed earlier, is obtained by scaling the quaternion so that its component is 1. For a 180° rotation around any axis, will be zero, which explains the Cayley limitation. The sum of the entries along the main diagonal (the trace), plus one, equals , which is . Thus we can write the trace itself as ; and from the previous version of the matrix we see that the diagonal entries themselves have the same form: , , and . So we can easily compare the magnitudes of all four quaternion components using the matrix diagonal. We can, in fact, obtain all four magnitudes using sums and square roots, and choose consistent signs using the skew-symmetric part of the off-diagonal entries: Alternatively, use a single square root and division This is numerically stable