Of the four matrices produced by this decomposition, the translation matrix, the matrix containing U, and the matrix containing V must all describe rigid-body tranformations, so the rigid-body component is effectively isolated in matrix S, which can only have elements along the diagonal. In general, the original matrix can only describe a global rescaling transformation if
where m is positive. The usual caveats regarding the final row of the original four-by-four matrix apply.
Aside from the added complication of a global rescaling component, the underlying geometry of the global rescaling model is quite similar to that of the rigid-body model. Schur decomposition can be used to define a new coordinate system in which all rotational components can be made parallel to either the xy or the yz plane. If rotations are parallel to the xy plane, an additional translation and rescaling occurs along the z-axis. Likewise, if rotations are parallel to the yz plane, additional translation and rescaling occur along the x-axis. For any magnification factor that is not unity, some point in space must always map to itself and an axis passing through this point can be viewed as the axis around which all rotations occur. This point can be identified by finding the real eigenvector of the original transformation that has a nonzero fourth element and rescaling that vector to make the fourth element equal to one.
In the numerical example, the point ( — 2.4515, — 2.6862, — 4.9621) is unchanged by the transformation. The other real eigenvector of the transformation,
identifies a vector parallel to the axis of rotation, so any point (-2.4515 + k*0.4347, -2.6862 + k*0.5382, -4.9621 +k*0.7221) lies on this rotational axis and maps to the point (-2.4515 + 2*k*0.4347, -2.6862 + 2*k*0.5382, - 4.9621 +2*k*0.7221).
The matrix logarithm provides an alternative to singular value decomposition for recognizing a global rescaling transformation and for decomposing the transformation into 5iOT«Zian«OM5 elementary transformations.
For the numerical example, the matrix logarithm is
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