v2010.10.26 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 377incomplete K ∈ C n×n , we might constrain F(U) thus:and in vector form, (42) (1787)ΘΦΘ ◦ F UF = K (843)δ(vec ΘΦΘ)(F ⊗F ) vec U = vec K (844)Because measurements K are complex, there are actually twice the numberof equality constraints as there are measurements.We can cut that number of constraints in half via vertical and horizontalmask Φ symmetry which forces the imaginary inverse transform to 0 : Theinverse subsampled transform in matrix form isand in vector formF H (ΘΦΘ ◦ F UF)F H = F H KF H (845)(F H ⊗F H )δ(vec ΘΦΘ)(F ⊗F ) vec U = (F H ⊗F H ) vec K (846)later abbreviatedP vec U = f (847)whereP (F H ⊗F H )δ(vec ΘΦΘ)(F ⊗F ) ∈ C n2 ×n 2 (848)Because of idempotence P = P 2 , P is a projection matrix. Because of itsHermitian symmetry [166, p.24]P = (F H ⊗F H )δ(vec ΘΦΘ)(F ⊗F ) = (F ⊗F ) H δ(vec ΘΦΘ)(F H ⊗F H ) H = P H(849)P is an orthogonal projector. 4.52 P vec U is real when P is real; id est, whenfor positive even integer n[Φ =]Φ 11 Φ(1, 2:n)Ξ∈ R n×n (850)ΞΦ(2:n, 1) ΞΦ(2:n, 2:n)Ξwhere Ξ∈ S n−1 is the order-reversing permutation matrix (1728). In words,this necessary and sufficient condition on Φ (for a real inverse subsampled