then so is . There are 3 roots, thus at least one of them must be purely real (+1 or −1). After recollection of these general facts from matrix theory, we return to the rotation matrix . It follows from its realness and orthogonality that we can find a such that: If a matrix can be found that gives the above form, and there is only one purely real component and it is −1, then we define R to be an improper rotation. Let us only consider the case, then, of matrices R that are proper rotations (the third eigenvalue is just 1). The third column of the matrix will then be equal to the invariant vector . Writing and for the first two columns of , this equation gives If has eigenvalue 1, then and has also eigenvalue 1, which implies that in that case . Finally, the matrix equation is transformed by means of a unitary matrix, which gives The columns of are orthonormal. The third column is still , the other two columns are perpendicular to . We can now see how our definition of improper rotation corresponds with the geometric interpretation: an improper rotation is a rotation around an axis (here, the axis corresponding to the third coordinate) and a reflection on a plane perpendicular to that axis. If we only restrict ourselves to matrices with determinant 1, we can thus see that they must be proper rotations. This result implies that any orthogonal matrix corresponding to a proper rotation is equivalent to a rotation over an angle around an axis . Equivalence classes The trace (sum of diagonal elements) of the real rotation matrix given above is . Since a trace is invariant under an orthogonal matrix similarity transformation, it follows that all matrices that are equivalent to by such orthogonal matrix transformations have the same trace: the trace is a class function. This matrix transformation is clearly an equivalence relation, that is, all such equivalent matrices form an equivalence class. In fact, all proper rotation rotation matrices form a group, usually denoted by SO(3) (the special orthogonal group in 3 dimensions) and all matrices with the same trace form an equivalence class in this group. All elements of such an equivalence class share their rotation angle, but all rotations are around different axes. If is an eigenvector of with eigenvalue 1, then is also