The orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices
Contents
where
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. The individual solution was later published. A proof is also given in
This problem is equivalent to finding the nearest orthogonal matrix to a given matrix
to write
Proof
One proof depends on basic properties of the standard 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 (i.e. orthogonal matrices with determinant 1, also known as special orthogonal matrices). In this case, one can write (using the above decomposition
where