v2009.01.01 - Convex Optimization

608 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS
where step function ψ is defined in (1465). In this circumstance,
S ⋆ = U A U H B = R ⋆T ∈ C n×n (1628)
optimal matrices (1619) now unitary are related by transposition.
optimal value of objective (1620) is
The
‖U A Σ A Q H A − S ⋆ U B Σ B Q H BR ⋆ ‖ F = ‖ |Υ| − |Λ| ‖ F (1629)
while the corresponding optimal value of trace maximization (1621) is
C.4.2.2
sup
R H =R −1
S H =S −1 Re tr(A T SBR) = tr(|Υ| |Λ|) (1630)
Diagonal matrices
Now suppose A and B are diagonal matrices
A = Υ = δ 2 (Υ) ∈ S n , δ(Υ) ∈ K M (1631)
B = Λ = δ 2 (Λ) ∈ S n , δ(Λ) ∈ K M (1632)
both having their respective main diagonal entries arranged in nonincreasing
order:
minimize ‖Υ − SΛR‖ F
R , S
subject to R H = R −1
(1633)
S H = S −1
Then we have a symmetric decomposition from unitary matrices as in (1625)
where
U A ∆ = √ δ(ψ(δ(Υ))) , Q A ∆ = √ δ(ψ(δ(Υ))) H , Σ A = |Υ| (1634)
U B ∆ = √ δ(ψ(δ(Λ))) , Q B ∆ = √ δ(ψ(δ(Λ))) H , Σ B = |Λ| (1635)
Procrustes solution (1619) again sees the transposition relationship
S ⋆ = U A U H B = R ⋆T ∈ C n×n (1628)
but both optimal unitary matrices are now themselves diagonal. So,
S ⋆ ΛR ⋆ = δ(ψ(δ(Υ)))Λδ(ψ(δ(Λ))) = δ(ψ(δ(Υ)))|Λ| (1636)