v2007.09.17 - Convex Optimization

C.4. TWO-SIDED ORTHOGONAL PROCRUSTES 549maximizes Re tr(A T SBR) : [236] [213] [38] [144] optimal orthogonal matricesS ⋆ = U A U H B ∈ R m×m , R ⋆ = Q B Q H A ∈ R n×n (1518)[sic] are not necessarily unique [150,7.4.13] because the feasible set is notconvex. The optimal value for the objective of minimization is, by (40)‖U A Σ A Q H A − S ⋆ U B Σ B Q H B R ⋆ ‖ F = ‖U A (Σ A − Σ B )Q H A ‖ F = ‖Σ A − Σ B ‖ F (1519)while the corresponding trace maximization has optimal value [31,III.6.12]sup | tr(A T SBR)| = sup Re tr(A T SBR) = Re tr(A T S ⋆ BR ⋆ ) = tr(ΣA TΣ B ) ≥ tr(AT B)R H =R −1R H =R −1S H =S −1 S H =S −1 (1520)for which it is necessaryA T S ⋆ BR ⋆ ≽ 0 , BR ⋆ A T S ⋆ ≽ 0 (1521)The lower bound on inner product of singular values in (1520) is due tovon Neumann. Equality is attained if U HA U B =I and QH B Q A =I .C.4.2.1Symmetric matricesNow optimizing over the complex manifold of unitary matrices (B.5.1),the upper bound on trace (1506) is thereby raised: Suppose we are givendiagonalizations for (real) symmetric A,B (A.5)A = W A ΥW TA ∈ S n , δ(Υ) ∈ K M (1522)B = W B ΛW T B ∈ S n , δ(Λ) ∈ K M (1523)having their respective eigenvalues in diagonal matrices Υ, Λ ∈ S n arrangedin nonincreasing order (membership to the monotone cone K M (377)). Thenby splitting eigenvalue signs, we invent a symmetric SVD-like decompositionA ∆ = U A Σ A Q H A ∈ S n , B ∆ = U B Σ B Q H B ∈ S n (1524)where U A , U B , Q A , Q B ∈ C n×n are unitary matrices defined by (conferA.6.5)U A ∆ = W A√δ(ψ(δ(Υ))) , QA ∆ = W A√δ(ψ(δ(Υ)))H, ΣA = |Υ| (1525)U B ∆ = W B√δ(ψ(δ(Λ))) , QB ∆ = W B√δ(ψ(δ(Λ)))H, ΣB = |Λ| (1526)