v2006.03.09 - Convex Optimization

E.7. ON VECTORIZED MATRICES OF HIGHER RANK 551E.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 ) (1554)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 (1551). 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(vecP) ”. 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 ,B 1 ,Z 1 ,A 2 ,B 2 ,Z 2 as defined for nonorthogonal projector (1447),and definingP 1 ∆ = A 1 A † 1 ∈ S m , P 2 ∆ = A 2 A † 2 ∈ S p (1555)where A 1 ∈ R m×n , Z 1 ∈ R m×k , A 2 ∈ R p×n , Z 2 ∈ R p×k , then given any X‖X −P 1 XP 2 ‖ F = inf ‖X −A 1 (A † 1+B 1 Z TB 1 , B 2 ∈R n×k 1 )X(A †T2 +Z 2 B2 T )A T 2 ‖ F (1556)