v2009.01.01 - Convex Optimization

602 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS
For A,B∈ S N whose eigenvalues λ(A), λ(B)∈ R N are respectively
arranged in nonincreasing order, and for nonincreasingly ordered
diagonalizations A = W A ΥWA T and B = W B ΛWB T [174] [206,2.1]
[207]
λ(A) T λ(B) = sup
U∈ R N×N
U T U=I
(confer (1612)) where optimal U is
tr(A T U T BU) ≥ tr(A T B) (1607)
U ⋆ = W B W T
A ∈ R N×N (1604)
We can push that upper bound higher using a result inC.4.2.1:
|λ(A)| T |λ(B)| = sup
U∈ C N×N
U H U=I
Re tr(A T U H BU) (1590)
For step function ψ as defined in (1465), optimal U becomes
U ⋆ = W B
√
δ(ψ(δ(Λ)))
H√
δ(ψ(δ(Υ)))W
T
A ∈ C N×N (1591)
C.3 Orthogonal Procrustes problem
Given matrices A,B∈ R n×N , their product having full singular value
decomposition (A.6.3)
AB T ∆ = UΣQ T ∈ R n×n (1592)
then an optimal solution R ⋆ to the orthogonal Procrustes problem
minimize ‖A − R T B‖ F
R
(1593)
subject to R T = R −1
maximizes tr(A T R T B) over the nonconvex manifold of orthogonal matrices:
[176,7.4.8]
R ⋆ = QU T ∈ R n×n (1594)
A necessary and sufficient condition for optimality
holds whenever R ⋆ is an orthogonal matrix. [138,4]
AB T R ⋆ ≽ 0 (1595)