v2009.01.01 - Convex Optimization

118 CHAPTER 2. CONVEX GEOMETRY
Because of a shortage of extreme directions, conic section (216) cannot
be hyperspherical by the extremes theorem (2.8.1.1.1).
2.9.2.5.2 Example. PSD cone inscription in three dimensions.
Theorem. Geršgorin discs. [176,6.1] [317]
For p∈R m + given A=[A ij ]∈ S m , then all eigenvalues of A belong to the union
of m closed intervals on the real line;
λ(A) ∈ m ⋃
i=1
⎧
⎪⎨
ξ ∈ R
⎪⎩
|ξ − A ii | ≤ ̺i
∆
= 1 m∑
p i
j=1
j ≠ i
⎫
⎪⎬ ⋃
p j |A ij | = m [A ii −̺i , A ii +̺i]
i=1 ⎪⎭
(220)
Furthermore, if a union of k of these m [intervals] forms a connected region
that is disjoint from all the remaining n −k [intervals], then there are
precisely k eigenvalues of A in this region.
⋄
To apply the theorem to determine positive semidefiniteness of symmetric
matrix A , we observe that for each i we must have
Suppose
A ii ≥ ̺i (221)
m = 2 (222)
so A ∈ S 2 . Vectorizing A as in (49), svec A belongs to isometrically
isomorphic R 3 . Then we have m2 m−1 = 4 inequalities, in the matrix entries
A ij with Geršgorin parameters p =[p i ]∈ R 2 + ,
p 1 A 11 ≥ ±p 2 A 12
p 2 A 22 ≥ ±p 1 A 12
(223)
which describe an intersection of four halfspaces in R m(m+1)/2 . That
intersection creates the proper polyhedral cone K (2.12.1) whose
construction is illustrated in Figure 42. Drawn truncated is the boundary
of the positive semidefinite cone svec S 2 + and the bounding hyperplanes
supporting K .
it is a rotation of the Lorentz cone in matrix dimension 2.