v2007.09.13 - Convex Optimization

108 CHAPTER 2. CONVEX GEOMETRY2.9.2.4.1 Example. Positive semidefinite matrix from extreme directions.Diagonalizability (A.5) of symmetric matrices yields the following results:Any symmetric positive semidefinite matrix (1249) can be written in theformA = ∑ λ i z i zi T = ÂÂT = ∑ â i â T i ≽ 0, λ ≽ 0 (197)iia conic combination of linearly independent extreme directions (â i â T i or z i z T iwhere ‖z i ‖=1), where λ is a vector of eigenvalues.If we limit consideration to all symmetric positive semidefinite matricesbounded such that trA=1C ∆ = {A ≽ 0 | trA = 1} (198)then any matrix A from that set may be expressed as a convex combinationof linearly independent extreme directions;A = ∑ iλ i z i z T i ∈ C , 1 T λ = 1, λ ≽ 0 (199)Implications are:1. set C is convex, (it is an intersection of PSD cone with hyperplane)2. because the set of eigenvalues corresponding to a given square matrix Ais unique, no single eigenvalue can exceed 1 ; id est, I ≽ A .Set C is an instance of Fantope (80).2.9.2.5 Positive semidefinite cone is generally not circularExtreme angle equation (196) suggests that the positive semidefinite conemight be invariant to rotation about its axis of revolution; id est, a circularcone. We investigate this now:2.9.2.5.1 Definition. Circular cone: 2.33a pointed closed convex cone having hyperspherical sections orthogonal toits axis of revolution about which the cone is invariant to rotation. △2.33 A circular cone is assumed convex throughout, although not so by other authors. Wealso assume a right circular cone.