v2009.01.01 - Convex Optimization

E.7. ON VECTORIZED MATRICES OF HIGHER RANK 669
E.6.4.3 PXP ≽ 0
In some circumstances, it may be desirable to limit the domain of test
y T Xy ≥ 0 for positive semidefiniteness; e.g., {‖y‖= 1}. Another example
of limiting domain-of-test is central to Euclidean distance geometry: For
R(V )= N(1 T ) , the test −V DV ≽ 0 determines whether D ∈ S N h is a
Euclidean distance matrix. The same test may be stated: For D ∈ S N h (and
optionally ‖y‖=1)
D ∈ EDM N ⇔ −y T Dy = 〈yy T , −D〉 ≥ 0 ∀y ∈ R(V ) (1867)
The test −V DV ≽ 0 is therefore equivalent to a test for nonnegativity of the
coefficient of orthogonal projection of −D on the range of each and every
vectorized extreme direction yy T from the positive semidefinite cone S N + such
that R(yy T )= R(y)⊆ R(V ). (The validity of this result is independent of
whether V is itself a projection matrix.)
E.7 on vectorized matrices of higher rank
E.7.1 PXP misinterpretation for higher-rank P
For a projection matrix P of rank greater than 1, PXP is generally not
commensurate with 〈P,X 〉 P as is the case for projector dyads (1864). 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 fallacy
is: vec PXP does not necessarily belong to the range of vectorized P ; the
most basic requirement for projection on R(vec P).
E.7.2
Orthogonal projection on matrix subspaces
With 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 as
defined for nonorthogonal projector (1762), and defining
then, given compatible X
P 1 ∆ = A 1 A † 1 ∈ S m , P 2 ∆ = A 2 A † 2 ∈ S p (1868)
‖X −P 1 XP 2 ‖ F = inf ‖X −A 1 (A † 1+B 1 Z T
B 1 , B 2 ∈R n×k 1 )X(A †T
2 +Z 2 B2 T )A T 2 ‖ F (1869)