v2006.03.09 - Convex Optimization

110 CHAPTER 2. CONVEX GEOMETRYThe only symmetric positive semidefinite matrix in S M +0-eigenvalues resides at the origin. (A.7.2.0.1)having M2.9.0.1 MembershipObserve the notation A ≽ 0 ; meaning, 2.30 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).2.9.0.1.1 Example. Equality constraints in semidefinite program (845).Employing properties of partial ordering (2.7.2.2) for the pointed closedconvex positive semidefinite cone, it is easy to show, given A + S = CS ≽ 0 ⇔ A ≼ C (157)2.9.1 PSD cone is convexThe set of all positive semidefinite matrices forms a convex cone in theambient space of symmetric matrices because any pair satisfies definition(138); [125,7.1] videlicet, for all ζ 1 , ζ 2 ≥ 0 and each and every A 1 , A 2 ∈ S Mζ 1 A 1 + ζ 2 A 2 ≽ 0 ⇐ A 1 ≽ 0, A 2 ≽ 0 (158)a fact easily verified by the definitive test for positive semidefiniteness of asymmetric matrix (A):A ≽ 0 ⇔ x T Ax ≥ 0 for each and every ‖x‖ = 1 (159)2.30 For matrices, notation A ≽B denotes comparison on S M with respect to the positivesemidefinite cone; (A.3.1) id est, A ≽B ⇔ A −B ∈ S M + , a generalization of comparisonon the real line. The symbol ≥ is reserved for scalar comparison on the real line R withrespect to the nonnegative real line R + as in a T y ≥ b, while a ≽ b denotes comparisonof vectors on R M with respect to the nonnegative orthant R M + .