v2007.11.26 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 1152.9.2.6 Boundary constituents of the positive semidefinite cone2.9.2.6.1 Lemma. Sum of positive semidefinite matrices.For A,B ∈ S M +rank(A + B) = rank(µA + (1 −µ)B) (216)over the open interval (0, 1) of µ .⋄Proof. Any positive semidefinite matrix belonging to the PSD conehas an eigen 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 . 2.9.2.6.2 Example. Rank function quasiconcavity. (confer3.3)For A,B ∈ R m×n [150,0.4]that follows from the fact [251,3.6]rankA + rankB ≥ rank(A + B) (217)dim R(A) + dim R(B) = dim R(A + B) + dim(R(A) ∩ R(B)) (218)For A,B ∈ S M +[46,3.4.2]rankA + rankB ≥ rank(A + B) ≥ min{rankA, rankB} (219)that follows from the factN(A + B) = N(A) ∩ N(B) , A,B ∈ S M + (134)Rank is a quasiconcave function on S M + because the right-hand inequality in(219) has the concave form (540); videlicet, Lemma 2.9.2.6.1. From this example we see, unlike convex functions, quasiconvex functionsare not necessarily continuous. (3.3) We also glean: