The orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices ${\displaystyle A}$ and ${\displaystyle B}$ and asked to find an orthogonal matrix ${\displaystyle \Omega }$ which most closely maps ${\displaystyle A}$ to ${\displaystyle B}$. Specifically, the orthogonal Procrustes problem is an optimization problem given by

$${\displaystyle {\begin{aligned}{\underset {\Omega }{\text{minimize}}}\quad &\|\Omega A-B\|_{F}\\{\text{subject to}}\quad &\Omega ^{T}\Omega =I,\end{aligned}}}$$

where ${\displaystyle \|\cdot \|_{F}}$ denotes the Frobenius norm. 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.

This problem is equivalent to finding the nearest orthogonal matrix to a given matrix ${\displaystyle M=BA^{T}}$, i.e. solving the closest orthogonal approximation problem

To find matrix ${\displaystyle R}$, one uses the singular value decomposition (for which the entries of ${\displaystyle \Sigma }$ are non-negative)

to write