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