31. The number of degrees of freedom of a rotation matrix is always less than the dimension of the matrix squared. 32. Here, we only describe the method based on the computation of the eigenvectors and eigenvalues of the rotation matrix . 33. However, it can also be represented by the 9 entries of a rotation matrix with 3 rows and 3 columns. 34. If the Jacobian matrix of the transformation is everywhere a scalar times a rotation matrix , then the transformation is conformal. 35. These singularities are not characteristic of the rotation matrix as such, and only occur with the usage of Euler angles. 36. Given a rotation matrix " M " the eigenvalues can calculated by solving the characteristic equation for that matrix. 37. To be a proper rotation matrix it must also satisfy \ det ( \ mathbf { R } ) = 1. 38. There are several methods to compute an axis and an angle from a rotation matrix ( see also axis-angle ). 39. It follows that a general rotation matrix in three dimensions has, up to a multiplicative constant, only one real eigenvector. 40. Thus we can extract from any 3? rotation matrix a rotation axis and an angle, and these completely determine the rotation.
