SO(n) is for each n a Lie group. It is compact and connected, but not simply connected. It is also a semi-simple group, in fact a simple group with the exception SO(4). The Lie algebra of SO(3) is denoted by and consists of all skew-symmetric 3 × 3 matrices. (The vector cross product can be expressed as the product of a skew-symmetric matrix and a vector). The most common basis for is
The matrices in the Lie algebra are not themselves rotations; the skew-symmetric matrices are derivatives. If then is in . Informally, an element of is the difference between the matrix of an infinitesimal rotation and the identity matrix, but "scaled up by a factor of infinity".
An actual "differential rotation", or infinitesimal rotation matrix has the form
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
(In 3 dimensions the trace of any rotation matrix must equal 1 + 2 cos(Angle) therefore the angle of rotation must be infinitesimal)
First, test the orthogonality condition, QTQ = I. The product is
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,
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,
Compare the products dAxdAy to dAydAx,
Since dθ dφ is second order, we discard it: thus, to first order, multiplication of infinitesimal rotation matrices is commutative. In fact,
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.
More intrinsically (i.e., without using coordinates), skew-symmetric linear transformations on a vector space V with an inner product may be defined as the bivectors on the space, which are sums of simple bivectors (2-blades) . The correspondence is given by the map where is the covector dual to the vector ; in orthonormal coordinates these are exactly the elementary skew-symmetric matrices. This characterization is used in interpreting the curl of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name.
The vector calculus operations of grad, curl, and div are most easily generalized and understood in the context of differential forms, which involves a number of steps. In a nutshell, they correspond to the derivatives of 0-forms, 1-forms, and 2-forms, respectively. The geometric interpretation of curl as rotation corresponds to identifying bivectors (2-vectors) in 3 dimensions with the special orthogonal Lie algebra (3) of infinitesimal rotations (in coordinates, skew-symmetric 3 × 3 matrices), while representing rotations by vectors corresponds to identifying 1-vectors (equivalently, 2-vectors) and (3), these all being 3-dimensional spaces.
Lie algebra R3
- See also Cross product
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 and consists of all skew-symmetric 3 × 3 matrices. This may be seen by differentiating the orthogonality condition, ATA = I, A ∈ SO(3).For an alternative derivation of , see Classical group.</ref> The Lie bracket of two elements of is, as for the Lie algebra of every matrix group, given by the matrix commutator, [A1, A2] = A1A2 − A2A1, which is again a skew-symmetric 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 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
for every vector x in R3.
This can be used to show that the Lie algebra (with commutator) is isomorphic to the Lie algebra R3 (with cross product). Under this isomorphism, an Euler vector corresponds to the linear map defined by .
In more detail, a most often suitable basis for as a 3-dimensional vector space is
The commutation relations of these basis elements are,
which agree with the relations of the three standard unit vectors of R3 under the cross product.
where a× is defined by:
One actually has
i.e., the commutator of skew-symmetric three-by-three matrices can be identified with the cross-product of three-vectors. Since the skew-symmetric three-by-three matrices are the Lie algebra of the rotation group this elucidates the relation between three-space , the cross product and three-dimensional rotations.
Skew-symmetric matrices over the field of real numbers form the tangent space to the real orthogonal group O(n) at the identity matrix; formally, the special orthogonal Lie algebra. In this sense, then, skew-symmetric matrices can be thought of as infinitesimal rotations.
Another way of saying this is that the space of skew-symmetric matrices forms the Lie algebra o(n) of the Lie group O(n). The Lie bracket on this space is given by the commutator:
It is easy to check that the commutator of two skew-symmetric matrices is again skew-symmetric:
The matrix exponential of a skew-symmetric matrix A is then an orthogonal matrix R:
The image of the exponential map of a Lie algebra always lies in the connected component of the Lie group that contains the identity element. In the case of the Lie group O(n), this connected component is the special orthogonal group SO(n), consisting of all orthogonal matrices with determinant 1. So R = exp(A) will have determinant +1. Moreover, since the exponential map of a connected compact Lie group is always surjective, it turns out that every orthogonal matrix with unit determinant can be written as the exponential of some skew-symmetric matrix. In the particular important case of dimension n = 2, the exponential representation for an orthogonal matrix reduces to the well-known polar form of a complex number of unit modulus. Indeed, if n=2, a special orthogonal matrix has the form
with a2 + b2 = 1. Therefore, putting a = cosθ and b = sin θ, it can be written
which corresponds exactly to the polar form cos θ + isin θ = eiθ of a complex number of unit modulus.
The exponential representation of an orthogonal matrix of order n can also be obtained starting from the fact that in dimension n any special orthogonal matrix R can be written as R = QSQT, where Q is orthogonal and S is a block diagonal matrix with blocks of order 2, plus one of order 1 if n is odd; since each single block of order 2 is also an orthogonal matrix, it admits an exponential form. Correspondingly, the matrix S writes as exponential of a skew-symmetric block matrix Σ of the form above, S = exp(Σ), so that R = Q exp(Σ)QT = exp(QΣQT), exponential of the skew-symmetric matrix QΣQT. Conversely, the surjectivity of the exponential map, together with the above-mentioned block-diagonalization for skew-symmetric matrices, implies the block-diagonalization for orthogonal matrices.
- Proposition 3.24
- Henning Makholm (https://math.stackexchange.com/users/14366/henning-makholm), If so(3) is the Lie algebra of SO(3) then why are the matrices of so(3) not rotation matrices?, URL (version: 2017-10-24): https://math.stackexchange.com/q/2488191
- (Goldstein, Poole & Safko 2002, §4.8)
- Proposition 3.24
- Shuangzhe Liu; Gõtz Trenkler (2008). "Hadamard, Khatri-Rao, Kronecker and other matrix products". Int J Information and systems sciences (Institute for scientific computing and education) 4 (1): 160–177. http://www.math.ualberta.ca/ijiss/SS-Volume-4-2008/No-1-08/SS-08-01-17.pdf.
- Infinitesimal transformation