v2007.11.26 - Convex Optimization

624 APPENDIX E. PROJECTIONThe test for positive semidefiniteness, then, is a test for nonnegativity ofthe coefficient of orthogonal projection of X on the range of each and everyvectorized extreme direction yy T (2.8.1) from the positive semidefinite conein the ambient space of symmetric matrices.E.6.4.3 PXP ≽ 0In some circumstances, it may be desirable to limit the domain of testy T Xy ≥ 0 for positive semidefiniteness; e.g., ‖y‖= 1. Another exampleof limiting domain-of-test is central to Euclidean distance geometry: ForR(V )= N(1 T ) , the test −V DV ≽ 0 determines whether D ∈ S N h is aEuclidean distance matrix. The same test may be stated: For D ∈ S N h (andoptionally ‖y‖=1)D ∈ EDM N ⇔ −y T Dy = 〈yy T , −D〉 ≥ 0 ∀y ∈ R(V ) (1785)The test −V DV ≽ 0 is therefore equivalent to a test for nonnegativity of thecoefficient of orthogonal projection of −D on the range of each and everyvectorized extreme direction yy T from the positive semidefinite cone S N + suchthat R(yy T ) = R(y) ⊆ R(V ). (The validity of this result is independent ofwhether V is itself a projection matrix.)E.7 on vectorized matrices of higher rankE.7.1 PXP misinterpretation for higher-rank PFor a projection matrix P of rank greater than 1, PXP is generally notcommensurate with 〈P,X 〉 P as is the case for projector dyads (1782). Yet〈P,P 〉for a symmetric idempotent matrix P of any rank we are tempted to say“ PXP is the orthogonal projection of X ∈ S m on R(vec P) ”. The fallacyis: vec PXP does not necessarily belong to the range of vectorized P ; themost basic requirement for projection on R(vec P) .E.7.2Orthogonal projection on matrix subspacesWith A 1 ∈ R m×n , B 1 ∈ R n×k , Z 1 ∈ R m×k , A 2 ∈ R p×n , B 2 ∈ R n×k , Z 2 ∈ R p×k asdefined for nonorthogonal projector (1680), and definingP 1 ∆ = A 1 A † 1 ∈ S m , P 2 ∆ = A 2 A † 2 ∈ S p (1786)