v2009.01.01 - Convex Optimization

114 CHAPTER 2. CONVEX GEOMETRY
For two-dimensional matrices (M =2, Figure 37)
{yy T ∈ S 2 | y ∈ R 2 } = ∂S 2 + (208)
while for one-dimensional matrices, in exception, (M =1,2.7)
{yy ∈ S | y≠0} = int S + (209)
Each and every extreme direction yy T makes the same angle with the
identity matrix in isomorphic R M(M+1)/2 , dependent only on dimension;
videlicet, 2.37
( )
(yy T 〈yy T , I〉 1
, I) = arccos = arccos √ ∀y ∈ R M (210)
‖yy T ‖ F ‖I‖ F M
2.9.2.4.1 Example. Positive semidefinite matrix from extreme directions.
Diagonalizability (A.5) of symmetric matrices yields the following results:
Any symmetric positive semidefinite matrix (1353) can be written in the
form
A = ∑ λ i z i zi T = ÂÂT = ∑ â i â T i ≽ 0, λ ≽ 0 (211)
i
i
a conic combination of linearly independent extreme directions (â i â T i or z i z T i
where ‖z i ‖=1), where λ is a vector of eigenvalues.
If we limit consideration to all symmetric positive semidefinite matrices
bounded such that trA=1
C ∆ = {A ≽ 0 | trA = 1} (212)
then any matrix A from that set may be expressed as a convex combination
of linearly independent extreme directions;
A = ∑ i
λ i z i z T i ∈ C , 1 T λ = 1, λ ≽ 0 (213)
Implications are:
1. set C is convex, (it is an intersection of PSD cone with hyperplane)
2. because the set of eigenvalues corresponding to a given square matrix A
is unique, no single eigenvalue can exceed 1 ; id est, I ≽ A .
Set C is an instance of Fantope (83).
2.37 Analogy with respect to the EDM cone is considered in [159, p.162] where it is found:
angle is not constant. Extreme directions of the EDM cone can be found in6.5.3.2. The
cone’s axis is −E =11 T − I (1000).