That is, any improper orthogonal 3x3 matrix may be decomposed as a proper rotation (from which an axis of rotation can be found as described above) followed by an inversion (multiplication by -1). It follows that the rotation axis of formula_7 is also the eigenvector of formula_24 corresponding to an eigenvalue of -1. As much as every tridimensional rotation has a rotation axis, also every tridimensional rotation has a plane, which is perpendicular to the rotation axis, and which is left invariant by the rotation. The rotation, restricted to this plane, is an ordinary 2D rotation. The proof proceeds similarly to the above discussion. First, suppose that all eigenvalues of the 3D rotation matrix A are real. This means that there is an orthogonal basis, made by the corresponding eigenvectors (which are necessarily orthogonal), over which the effect of the rotation matrix is just stretching it. If we write A in this basis, it is diagonal; but a diagonal orthogonal matrix is made of just +1's and -1's in the diagonal entries. Therefore, we don't have a proper rotation, but either the identity or the result of a sequence of reflections. It follows, then, that a proper rotation has some complex eigenvalue. Let v be the corresponding eigenvector. Then, as we showed in the previous topic, formula_29 is also an eigenvector, and formula_30 and formula_31 are such that their scalar product vanishes:  because, since formula_33 is real, it equals its complex conjugate formula_34, and formula_35 and formula_36 are both representations of the same scalar product between formula_37 and formula_29. This means formula_30 and formula_31 are orthogonal vectors. Also, they are both real vectors by construction. These vectors span the same subspace as formula_37 and formula_29, which is an invariant subspace under the application of A. Therefore, they span an invariant plane. This plane is orthogonal to the invariant axis, which corresponds to the remaining eigenvector of A, with eigenvalue 1, because of the orthogonality of the eigenvectors of A.       3-sphere  In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point.