Generalized/constrained Procrustes Problems
There are a number of related problems to the classical orthogonal Procrustes problem. One might generalize it by seeking the closest matrix in which the columns are orthogonal, but not necessarily orthonormal.
Alternately, one might constrain it by only allowing rotation matrices (i.e. orthogonal matrices with determinant 1, also known as special orthogonal matrices). In this case, one can write (using the above decomposition )
where is a modified, with the smallest singular value replaced by (+1 or -1), and the other singular values replaced by 1, so that the determinant of R is guaranteed to be positive. For more information, see the Kabsch algorithm.
Read more about this topic: Orthogonal Procrustes Problem
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