v2009.01.01 - Convex Optimization

Appendix A
Linear algebra
A.1 Main-diagonal δ operator, λ , trace, vec
We introduce notation δ denoting the main-diagonal linear self-adjoint
operator. 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 in
the natural order;
δ(A) ∈ R N (1318)
Operating on a vector y ∈ R N , δ naturally returns a diagonal matrix;
δ(y) ∈ S N (1319)
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 (1320)
Defined in this manner, main-diagonal linear operator δ is self-adjoint
[197,3.10,9.5-1]; A.1 videlicet, (2.2)
δ(A) T y = 〈δ(A), y〉 = 〈A , δ(y)〉 = tr ( A T δ(y) ) (1321)
A.1 Linear operator T : R m×n → R M×N is self-adjoint when, for each and every
X 1 , X 2 ∈ R m×n 〈T(X 1 ), X 2 〉 = 〈X 1 , T(X 2 )〉
2001 Jon Dattorro. CO&EDG version 2009.01.01. All rights reserved.
Citation: Jon Dattorro, Convex Optimization & Euclidean Distance Geometry,
Meboo Publishing USA, 2005.
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