Orthogonal Procrustes problem

The orthogonal Procrustes problem^[1] 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 ${\displaystyle B}$.^[2][3] 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.