v2007.09.13 - Convex Optimization

544 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTSC.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 (1495) makes 1 T of compatible dimension;R ⋆T [C −B11 T 1 NBV ] + A11 T 1 N(1496)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.5.2)A ∆ = Q A Λ A Q T A , B ∆ = Q B Λ B Q T B (1497)where eigenvalues are arranged in their respective diagonal matrix Λ innonincreasing order, then an optimal solution [85]to the two-sided orthogonal Procrustes problemR ⋆ = Q B Q T A ∈ R N×N (1498)minimize ‖A − R T BR‖ FR= minimize tr ( A T A − 2A T R T BR + B T B )subject to R T = R −1 Rsubject to R T = R −1 (1499)maximizes tr(A T R T BR) over the nonconvex manifold of orthogonal matrices.Optimal product R ⋆T BR ⋆ has the eigenvectors of A but the eigenvalues of B .[113,7.5.1] The optimal value for the objective of minimization is, by (40)‖Q A Λ A Q T A −R ⋆T Q B Λ B Q T BR ⋆ ‖ F = ‖Q A (Λ A −Λ B )Q T A ‖ F = ‖Λ A −Λ B ‖ F (1500)while the corresponding trace maximization has optimal valuesup tr(A T R T BR) = tr(A T R ⋆T BR ⋆ ) = tr(Λ A Λ B ) ≥ tr(A T B) (1501)R T =R −1