v2010.10.26 - Convex Optimization

C.4. TWO-SIDED ORTHOGONAL PROCRUSTES 663where V ∈ S N is the geometric centering matrix (B.4.1). Now we define thefull singular value decompositionand an optimal rotation matrixAV B T UΣQ T ∈ R n×n (1720)R ⋆ = QU T ∈ R n×n (1713)The desired result is an optimally rotated offset listR ⋆T BV + A(I − V ) ≈ A (1721)which most closely matches the list in A . Equality is attained when the listsare precisely related by a rotation/reflection and an offset. When R ⋆T B=Aor B1=A1=0, this result (1721) reduces to R ⋆T B ≈ A .C.3.1.1Translation of extended listSuppose an optimal rotation matrix R ⋆ ∈ R n×n were derived as beforefrom matrix B ∈ R n×N , but B is part of a larger list in the columns of[C B ]∈ R n×M+N where C ∈ R n×M . In that event, we wish to applythe rotation/reflection and translation to the larger list. The expressionsupplanting the approximation in (1721) makes 1 T of compatible dimension;R ⋆T [C −B11 T 1 NBV ] + A11 T 1 N(1722)id est, C −B11 T 1 N ∈ Rn×M and A11 T 1 N ∈ Rn×M+N .C.4 Two-sided orthogonal ProcrustesC.4.0.1MinimizationGiven symmetric A,B∈ S N , each having diagonalization (A.5.1)A Q A Λ A Q T A , B Q B Λ B Q T B (1723)where eigenvalues are arranged in their respective diagonal matrix Λ innonincreasing order, then an optimal solution [130]R ⋆ = Q B Q T A ∈ R N×N (1724)