v2007.09.13 - Convex Optimization

98 CHAPTER 2. CONVEX GEOMETRYThe positive definite (full-rank) matrices comprise the cone interior, 2.30int S M + = { A ∈ S M | A ≻ 0 }= { A ∈ S M | y T Ay>0 ∀ ‖y‖ = 1 }= {A ∈ S M + | rankA = M}(161)while all singular positive semidefinite matrices (having at least one 0eigenvalue) reside on the cone boundary (Figure 31); (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 } (162)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 M2.9.0.1 MembershipObserve the notation A ≽ 0 ; meaning, 2.31 matrix A is symmetricand belongs to the positive semidefinite cone in the subspace ofsymmetric matrices, whereas A ≻ 0 denotes membership to that cone’sinterior. (2.13.2) This notation further implies that coordinates [sic] fororthogonal expansion of a positive (semi)definite matrix must be its(nonnegative) positive eigenvalues (2.13.7.1.1,E.6.4.1.1) when expandedin its eigenmatrices (A.5.1).Generalizing comparison on the real line, the notation A ≽B denotescomparison with respect to the positive semidefinite cone; (A.3.1) id est,A ≽B ⇔ A −B ∈ S M + but neither matrix A or B necessarily belongs tothe positive semidefinite cone. Yet, (1282) A ≽B , B ≽0 ⇒ A≽0 ; id est,A ∈ S M + .2.30 The remaining inequalities in (160) also become strict for membership to the coneinterior.2.31 The symbol ≥ is reserved for scalar comparison on the real line R with respect to thenonnegative real line R + as in a T y ≥ b, while a ≽ b denotes comparison of vectors onR M with respect to the nonnegative orthant R M + (2.3.1.1).