v2010.10.26 - Convex Optimization

Appendix ALinear algebraA.1 Main-diagonal δ operator, λ , trace, vecWe introduce notation δ denoting the main-diagonal linear self-adjointoperator. When linear function δ operates on a square matrix A∈ R N×N ,δ(A) returns a vector composed of all the entries from the main diagonal inthe natural order;δ(A) ∈ R N (1415)Operating on a vector y ∈ R N , δ naturally returns a diagonal matrix;δ(y) ∈ S N (1416)Operating recursively on a vector Λ∈ R N or diagonal matrix Λ∈ S N ,δ(δ(Λ)) returns Λ itself;δ 2 (Λ) ≡ δ(δ(Λ)) Λ (1417)Defined in this manner, main-diagonal linear operator δ is self-adjoint[227,3.10,9.5-1]; A.1 videlicet, (2.2)δ(A) T y = 〈δ(A), y〉 = 〈A , δ(y)〉 = tr ( A T δ(y) ) (1418)A.1 Linear operator T : R m×n → R M×N is self-adjoint when, ∀ X 1 , X 2 ∈ R m×n〈T(X 1 ), X 2 〉 = 〈X 1 , T(X 2 )〉2001 Jon Dattorro. co&edg version 2010.10.26. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.10.26.591