v2009.01.01 - Convex Optimization

A.1. MAIN-DIAGONAL δ OPERATOR, λ , TRACE, VEC 541
A.1.2
Majorization
A.1.2.0.1 Theorem. (Schur) Majorization. [344,7.4] [176,4.3]
[177,5.5] Let λ∈ R N denote a given vector of eigenvalues and let
δ ∈ R N denote a given vector of main diagonal entries, both arranged in
nonincreasing order. Then
and conversely
∃A∈ S N λ(A)=λ and δ(A)= δ ⇐ λ − δ ∈ K ∗ λδ (1327)
A∈ S N ⇒ λ(A) − δ(A) ∈ K ∗ λδ (1328)
The difference belongs to the pointed polyhedral cone of majorization (empty
interior, confer (283))
K ∗ λδ
∆
= K ∗ M+ ∩ {ζ1 | ζ ∈ R} ∗ (1329)
where K ∗ M+ is the dual monotone nonnegative cone (389), and where the
dual of the line is a hyperplane; ∂H = {ζ1 | ζ ∈ R} ∗ = 1 ⊥ .
⋄
Majorization cone K ∗ λδ is naturally consequent to the definition of
majorization; id est, vector y ∈ R N majorizes vector x if and only if
k∑
x i ≤
i=1
k∑
y i ∀ 1 ≤ k ≤ N (1330)
i=1
and
1 T x = 1 T y (1331)
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 nonincreasing
order. Then
∃A∈ S N h λ(A)=λ ⇐ λ ∈ K ∗ λδ (1332)
and conversely
where K ∗ λδ
is defined in (1329).
⋄
A∈ S N h ⇒ λ(A) ∈ K ∗ λδ (1333)