Orthogonal Procrustes problem

The orthogonal Procrustes problem is a <a href="/facts/Matrix_approximation/8t3v6wk3">matrix approximation</a> problem in <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>. In its classical form, one is given two <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
 and 
 
 
 
 B
 
 
 {\displaystyle B}
 
 and asked to find an <a href="/facts/Orthogonal_matrix/WDXBXYB5">orthogonal matrix</a> 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 which most closely maps 
 
 
 
 A
 
 
 {\displaystyle A}
 
 to 
 
 
 
 B
 
 
 {\displaystyle B}
 
. Specifically, the orthogonal Procrustes problem is an <a href="/facts/Optimization_problem/QzwJfFEE">optimization problem</a> given by

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

where 
 
 
 
 ‖
 ⋅
 
 ‖
 
 F
 
 
 
 
 {\displaystyle \|\cdot \|_{F}}
 
 denotes the <a href="/facts/Frobenius_norm/UJvsb9fF">Frobenius norm</a>. This is a special case of <a href="/facts/Wahba%2527s_problem/qbJVLbJE">Wahba's problem</a> (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 <a href="/facts/Procrustes/wurg52pt">Procrustes</a> refers to a bandit from Greek mythology who made his victims fit his bed by either stretching their limbs or cutting them off.

Orthogonal Procrustes problem open-in-new

Orthogonal Procrustes problem