Of the five matrices in a decomposition such as the one just illustrated, the matrix containing U, the matrix containing V

and the two translational matrices are all guaranteed to describe rigid-body transformations. Consequently, if the original matrix describes a rigid-body transformation, the matrix containing S must also describe a rigid-body transformation. Since the singular value decomposition ensures that all the nondiagonal elements of S are zero, the matrix containing S can only describe a rigid-body transformation if

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