v2007.09.13 - Convex Optimization

482 APPENDIX A. LINEAR ALGEBRAA.1.2MajorizationA.1.2.0.1 Theorem. (Schur) Majorization. [298,7.4] [149,4.3][150,5.5] Let λ∈ R N denote a given vector of eigenvalues and letδ ∈ R N denote a given vector of main diagonal entries, both arranged innonincreasing order. Thenand conversely∃A∈ S N λ(A)=λ and δ(A)= δ ⇐ λ − δ ∈ K ∗ λδ (1223)A∈ S N ⇒ λ(A) − δ(A) ∈ K ∗ λδ (1224)The difference belongs to the pointed polyhedral cone of majorization (emptyinterior, confer (271))K ∗ λδ∆= K ∗ M+ ∩ {ζ1 | ζ ∈ R} ∗ (1225)where K ∗ M+ is the dual monotone nonnegative cone (376), and where thedual of the line is a hyperplane; ∂H = {ζ1 | ζ ∈ R} ∗ = 1 ⊥ .⋄Majorization cone K ∗ λδ is naturally consequent to the definition ofmajorization; id est, vector y ∈ R N majorizes vector x if and only ifk∑x i ≤i=1k∑y i ∀ 1 ≤ k ≤ N (1226)i=1and1 T x = 1 T y (1227)Under these circumstances, rather, vector x is majorized by vector y .In the particular circumstance δ(A)=0, we get:A.1.2.0.2 Corollary. Symmetric hollow majorization.Let λ∈ R N denote a given vector of eigenvalues arranged in nonincreasingorder. Then∃A∈ S N h λ(A)=λ ⇐ λ ∈ K ∗ λδ (1228)and converselywhere K ∗ λδis defined in (1225).⋄A∈ S N h ⇒ λ(A) ∈ K ∗ λδ (1229)