v2009.01.01 - Convex Optimization

104 CHAPTER 2. CONVEX GEOMETRY
2.9 Positive semidefinite (PSD) cone
The cone of positive semidefinite matrices studied in this section
is arguably the most important of all non-polyhedral cones whose
facial structure we completely understand.
−Alexander Barvinok [25, p.78]
2.9.0.0.1 Definition. Positive semidefinite cone.
The set of all symmetric positive semidefinite matrices of particular
dimension M is called the positive semidefinite cone:
S M ∆
+ = { A ∈ S M | A ≽ 0 }
= { A ∈ S M | y T Ay ≥0 ∀ ‖y‖ = 1 }
= ⋂ {
A ∈ S M | 〈yy T , A〉 ≥ 0 } (173)
‖y‖=1
formed by the intersection of an infinite number of halfspaces (2.4.1.1) in
vectorized variable A , 2.35 each halfspace having partial boundary containing
the origin in isomorphic R M(M+1)/2 . It is a unique immutable proper cone
in the ambient space of symmetric matrices S M .
The positive definite (full-rank) matrices comprise the cone interior
int S M + = { A ∈ S M | A ≻ 0 }
= { A ∈ S M | y T Ay>0 ∀ ‖y‖ = 1 }
= ⋂ {
A ∈ S M | 〈yy T , A〉 > 0 }
‖y‖=1
= {A ∈ S M + | rankA = M}
(174)
while all singular positive semidefinite matrices (having at least one
0 eigenvalue) reside on the cone boundary (Figure 37); (A.7.5)
∂S M + = { A ∈ S M | min{λ(A) i , i=1... M} = 0 }
= { A ∈ S M + | 〈yy T , A〉=0 for some ‖y‖ = 1 }
= { A ∈ S M + | rankA < M } (175)
where λ(A)∈ R M holds the eigenvalues of A .
△
2.35 infinite in number when M >1. Because y T A y=y T A T y , matrix A is almost always
assumed symmetric. (A.2.1)