v2009.01.01 - Convex Optimization

C.4. TWO-SIDED ORTHOGONAL PROCRUSTES 607
maximizes Re tr(A T SBR) : [272] [250] [45] [169] optimal orthogonal matrices
S ⋆ = U A U H B ∈ R m×m , R ⋆ = Q B Q H A ∈ R n×n (1619)
[sic] are not necessarily unique [176,7.4.13] because the feasible set is not
convex. The optimal value for the objective of minimization is, by (41)
‖U A Σ A Q H A − S ⋆ U B Σ B Q H BR ⋆ ‖ F = ‖U A (Σ A − Σ B )Q H A ‖ F = ‖Σ A − Σ B ‖ F (1620)
while the corresponding trace maximization has optimal value [38,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 −1
R H =R −1
S H =S −1 S H =S −1 (1621)
for which it is necessary
A T S ⋆ BR ⋆ ≽ 0 , BR ⋆ A T S ⋆ ≽ 0 (1622)
The lower bound on inner product of singular values in (1621) is due to
von Neumann. Equality is attained if U H
A U B =I and QH B Q A =I .
C.4.2.1
Symmetric matrices
Now optimizing over the complex manifold of unitary matrices (B.5.1),
the upper bound on trace (1607) is thereby raised: Suppose we are given
diagonalizations for (real) symmetric A,B (A.5)
A = W A ΥW T
A ∈ S n , δ(Υ) ∈ K M (1623)
B = W B ΛW T B ∈ S n , δ(Λ) ∈ K M (1624)
having their respective eigenvalues in diagonal matrices Υ, Λ ∈ S n arranged
in nonincreasing order (membership to the monotone cone K M (390)). Then
by splitting eigenvalue signs, we invent a symmetric SVD-like decomposition
A ∆ = U A Σ A Q H A ∈ S n , B ∆ = U B Σ B Q H B ∈ S n (1625)
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| (1626)
U B ∆ = W B
√
δ(ψ(δ(Λ))) , QB ∆ = W B
√
δ(ψ(δ(Λ)))
H
, ΣB = |Λ| (1627)