The orthogonal Procrustes problem[1] is a matrix approximation problem in linear algebra.
and asked to find an orthogonal matrix
This is a special case of Wahba's problem (with identical weights; instead of considering two matrices, in Wahba's problem the columns of the matrices are considered as individual vectors).
Another difference is that Wahba's problem tries to find a proper rotation matrix instead of just an orthogonal one.
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.
This problem was originally solved by Peter Schönemann in a 1964 thesis, and shortly after appeared in the journal Psychometrika.
, i.e. solving the closest orthogonal approximation problem To find matrix
, one uses the singular value decomposition (for which the entries of
are non-negative) to write One proof depends on the basic properties of the Frobenius inner product that induces the Frobenius norm: where
One might generalize it by seeking the closest matrix in which the columns are orthogonal, but not necessarily orthonormal.
[5] Alternately, one might constrain it by only allowing rotation matrices (i.e. orthogonal matrices with determinant 1, also known as special orthogonal matrices).
, 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.
The unbalanced Procrustes problem concerns minimizing the norm of
, or alternately with complex valued matrices.
This is a problem over the Stiefel manifold
To distinguish, the standard Procrustes problem (
) is referred to as the balanced problem in these contexts.