v2010.10.26 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 135rank-one matrices); its geometry becomes intricate. How all the extremedirections can be swept by an inscribed polyhedral cone, 2.49 similarly to theforegoing example, remains an open question.2.9.2.8.4 Exercise. Dual inscription.Find dual proper polyhedral cone K ∗ from Figure 48.2.9.2.9 Boundary constituents of the positive semidefinite cone2.9.2.9.1 Lemma. Sum of positive semidefinite matrices.For A,B ∈ S M +rank(A + B) = rank(µA + (1 −µ)B) (257)over open interval (0, 1) of µ .Proof. Any positive semidefinite matrix belonging to the PSD conehas an eigenvalue decomposition that is a positively scaled sum of linearlyindependent symmetric dyads. By the linearly independent dyads definitioninB.1.1.0.1, rank of the sum A +B is equivalent to the number of linearlyindependent dyads constituting it. Linear independence is insensitive tofurther positive scaling by µ . The assumption of positive semidefinitenessprevents annihilation of any dyad from the sum A +B . 2.9.2.9.2 Example. Rank function quasiconcavity. (confer3.8)For A,B ∈ R m×n [202,0.4]that follows from the fact [331,3.6]rankA + rankB ≥ rank(A + B) (258)dim R(A) + dim R(B) = dim R(A + B) + dim(R(A) ∩ R(B)) (259)For A,B ∈ S M +rankA + rankB ≥ rank(A + B) ≥ min{rankA, rankB} (1466)that follows from the factN(A + B) = N(A) ∩ N(B) , A,B ∈ S M + (160)2.49 It is not necessary to sweep the entire boundary in higher dimension.⋄