v2010.10.26 - Convex Optimization

116 CHAPTER 2. CONVEX GEOMETRY2.9.0.1 MembershipObserve notation A ≽ 0 denoting a positive semidefinite matrix; 2.38 meaning(confer2.3.1.1), matrix A belongs to the positive semidefinite cone in thesubspace of symmetric matrices whereas A ≻ 0 denotes membership to thatcone’s interior. (2.13.2) Notation A ≻ 0 , denoting a positive definite matrix,can be read: symmetric matrix A is greater than the origin with respect to thepositive semidefinite cone. These notations further imply that coordinates[sic] for orthogonal 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.5.0.3); id est, eigenvalues must be (nonnegative)positive.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, (1486) A ≽B , B ≽0 ⇒ A≽0 ; id est,A∈ S M + . (confer Figure 63)2.9.0.1.1 Example. Equality constraints in semidefinite program (649).Employing properties of partial order (2.7.2.2) for the pointed closed convexpositive semidefinite cone, it is easy to show, given A + S = CS ≽ 0 ⇔ A ≼ CS ≻ 0 ⇔ A ≺ C(194)2.9.1 Positive semidefinite cone is convexThe set of all positive semidefinite matrices forms a convex cone in theambient space of symmetric matrices because any pair satisfies definition(175); [202,7.1] 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 (195)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 (196)2.38 the same as nonnegative definite matrix.