v2010.10.26 - Convex Optimization

126 CHAPTER 2. CONVEX GEOMETRYwe may ignore the fact that vectorized nullspace svec basis N(A i ) is a propersubspace of the hyperplane. We may think instead in terms of wholehyperplanes because equivalence (1590) says that the positive semidefinitecone effectively filters that subset of the hyperplane, whose normal is A i ,that constitutes N(A i ).And so hyperplane intersection makes a line intersecting the positivesemidefinite cone but only at the origin. In this hypothetical example,smallest face containing those two matrices therefore comprises the entirecone because every positive semidefinite matrix has nullspace containing 0.The smaller the intersection, the larger the smallest face. 2.9.2.5.2 Exercise. Disparate elements.Prove that (230) holds for an arbitrary set {A i ∈ S M + }. One way is by showing⋂ N(Ai ) ∩ S M + = conv({A i }) ⊥ ∩ S M + ; with perpendicularity ⊥ as in (373). 2.432.9.2.6 face of all PSD matrices having same principal submatrixNow we ask what is the smallest face of the positive semidefinite conecontaining all matrices having a principal submatrix in common; in otherwords, that face containing all PSD matrices (of any rank) with particularentries fixed − the smallest face containing all PSD matrices whose fixedentries correspond to some given principal submatrix ΦAΦ . To maintaingenerality, 2.44 we move an extracted principal submatrix ˆΦ T A ˆΦ∈ S rankΦ+ intoleading position via permutation Ξ from (228): for A∈ S M +[ΞAΞ T ˆΦT A ˆΦ BB T C]∈ S M + (231)By properties of partitioned PSD matrices inA.4.0.1,([basis NˆΦT A ˆΦ]) [ ]B 0B T ⊇C I − CC † (232)[ ] 0Hence N(ΞXΞ T ) ⊇ N(ΞAΞ T ) span in a smallest face F formulaI2.45because all PSD matrices, given fixed principal submatrix, are admitted:2.43 Hint: (1590) (1911).2.44 to fix any principal submatrix; not only leading principal submatrices.2.45 meaning, more pertinently, I − Φ is dropped from (227).