Orthogonal Procrustes problem

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

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

where $\|\cdot \|_{F}$ 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.

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