2) Reverse rotate the axis-point pair which was done in step 1 (undoing step 1) This can be written more concisely as where is the cross product matrix of ; the expression is the outer product, and is the identity matrix. Alternatively, the matrix entries are: where is the Levi-Civita symbol with . This is a matrix form of Rodrigues' rotation formula, (or the equivalent, differently parametrized Euler–Rodrigues formula) with In the rotation of a vector around the axis by an angle can be written as: If the 3D space is right-handed and , this rotation will be counterclockwise when points towards the observer (Right-hand rule). Explicitly, with a right-handed orthonormal basis, Note the striking merely apparent differences to the equivalent Lie-algebraic formulation below. Properties For any -dimensional rotation matrix acting on (The rotation is an orthogonal matrix) It follows that: A rotation is termed proper if , and improper (or a roto-reflection) if . For even dimensions , the eigenvalues of a proper rotation occur as pairs of complex conjugates which are roots of unity: for , which is real only for . Therefore, there may be no vectors fixed by the rotation (), and thus no axis of rotation. Any fixed eigenvectors occur in pairs, and the axis of rotation is an even-dimensional subspace. For odd dimensions , a proper rotation will have an odd number of eigenvalues, with at least one and the axis of rotation will be an odd dimensional subspace. Proof: Here is the identity matrix, and we use , as well as since is odd. Therefore, , meaning there is a null vector with , that is , a fixed eigenvector. There may also be pairs of fixed eigenvectors in the even-dimensional subspace orthogonal to , so the total dimension of fixed eigenvectors is odd. For example, in 2-space , a rotation by angle has eigenvalues and , so there is no axis of rotation except when , the case of the null rotation. In 3-space , the axis of a non-null proper rotation is always a unique line, and a rotation around this axis by angle has eigenvalues . In 4-space , the four eigenvalues are of the form . The null rotation has . The case of is called a simple rotation, with two unit eigenvalues forming an axis plane, and a two-dimensional rotation orthogonal to the axis plane. Otherwise, there is