When the two-by-two submatrix has the form a b — b a the simplest way to confirm that a rigid-body transformation is described is to compute the determinant of the matrix, a2 + b2, and confirm that it is equal to 1. In the three-dimensional case to be described later, simple inspection for symmetry and computation of the determinant are not sufficient. A more general way to determine whether a transformation matrix describes a rigid-body transformation is to use singular value decomposition of the two-by-two submatrix. Any real two-by-two matrix can be decomposed into the product of three two-by-two matrices, U, S, and V that have special properties. The U and V matrices have the property that they are orthonormal, which means that they can be viewed as describing rigid-body rotations. The S matrix has the property that all of its offdiagonal elements are zero. The product U * S * V' gives the original matrix that was decomposed.

For the two-by-two submatrix of the example described earlier, the singular value decomposition gives

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