The matrix S always contains the three scaling factors along the diagonal. However, their ordering along the diagonal alone cannot be used to determine which axis was rescaled by a given value. This must be deduced by considering their ordering in the context of the U and V transformations. The U and V matrices will also reveal whether the rescaling occurs before or after rotation. If rescaling occurs first, matrix V will have only three nonzero elements, one in each row and one in each column. Furthermore, these nonzero elements will all have absolute values of unity. If rescaling occurs after rotation, matrix U will have these properties instead. Since the rigid-body model and global rescaling model are both special cases of the nine-parameter affine model, the matrix S must also be inspected if these special cases are to be excluded.
Computation of eigenvectors, eigenvalues, Schur decomposition, or matrix logarithms of nine-parameter spatial transformations do not reveal any special properties or geometric interpretations that do not also apply to the general 12-parameter affine model, so discussion of these issues is deferred to the next section.
For the second numerical example, singular value decomposition gives
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