Orthogonal Procrustes problem


The orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices and and asked to find an orthogonal matrix which most closely maps to. Specifically,
where denotes the Frobenius norm. This is a special case of Wahba's problem.
The name Procrustes refers to a bandit from Greek mythology who made his victims fit his bed by either stretching their limbs or cutting them off.

Solution

This problem was originally solved by Peter Schönemann in a 1964 thesis, and shortly after appeared in the journal Psychometrika. A proof appeared in 1998.
This problem is equivalent to finding the nearest orthogonal matrix to a given matrix. To find this orthogonal matrix, one uses the singular value decomposition
to write

Proof

One proof depends on basic properties of the matrix inner product that induces the Frobenius norm:

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. In this case, one can write
where is a modified, with the smallest singular value replaced by , 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.