v2010.10.26 - Convex Optimization

114 CHAPTER 2. CONVEX GEOMETRY2.9.0.0.1 Definition. Positive semidefinite cone.The set of all symmetric positive semidefinite matrices of particulardimension 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 }(191)‖y‖=1= {A ∈ S M + | rankA ≤ M }formed by the intersection of an infinite number of halfspaces (2.4.1.1) invectorized variable 2.37 A , each halfspace having partial boundary containingthe origin in isomorphic R M(M+1)/2 . It is a unique immutable proper conein the ambient space of symmetric matrices S M .The positive definite (full-rank) matrices comprise the cone interiorint 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 }(192)while all singular positive semidefinite matrices (having at least one0 eigenvalue) reside on the cone boundary (Figure 43); (A.7.5)∂S M + = { A ∈ S M | A ≽ 0, A ⊁ 0 }= { 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 }(193)where λ(A)∈ R M holds the eigenvalues of A .The only symmetric positive semidefinite matrix in S M +0-eigenvalues resides at the origin. (A.7.3.0.1)△having M2.37 infinite in number when M >1. Because y T A y=y T A T y , matrix A is almost alwaysassumed symmetric. (A.2.1)