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For all rigid-body transformations, the matrix U must not only be orthonormal (as guaranteed by the Schur decomposition), but can also be constrained to have all nondiagonal elements of the fourth row and fourth column equal to zero and the corresponding diagonal element in the fourth row and fourth column equal to one. As a result, the matrix U' can be viewed as simply defining a new set of coordinate axes with the same origin as the original set. The transformation V is applied in this new coordinate system and then the matrix U reverts the

As a result, V can be viewed as a two-dimensional rigid-body motion constrained to the new xy plane defined by U' and a translation along the new z-axis defined by U'. Note that the order of these two transformations does not matter. It should be noted that routines to perform Schur decomposition will not necessarily automatically produce a V matrix in this specific format, but methods are available to ensure that this is the case [1, 3].

For the numerical example:

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