v2009.01.01 - Convex Optimization

E.6. VECTORIZATION INTERPRETATION, 665
matrices P 1 and P 2 we rewrite (1851):
y T Xz
y T y z T z yzT =
yyT
y T y X zzT
z T z
∆
= P 1 XP 2 (1852)
So for projector dyads, projection (1852) is the orthogonal projection in R mn
if and only if projectors P 1 and P 2 are symmetric; E.10 in other words,
for orthogonal projection on the range of a vectorized dyad yz T , the
term outside the vector inner-products 〈 〉 in (1851) must be identical
to the terms inside in three places.
When P 1 and P 2 are rank-one symmetric projectors as in (1852), (30)
R(vec P 1 XP 2 ) = R(vecyz T ) in R mn (1853)
and
P 1 XP 2 − X ⊥ yz T in R mn (1854)
When y=z then P 1 =P 2 =P T 2 and
P 1 XP 1 = 〈P 1 , X〉 P 1 = 〈P 1 , X〉
〈P 1 , P 1 〉 P 1 (1855)
meaning, P 1 XP 1 is equivalent to orthogonal projection of matrix X on
the range of vectorized projector dyad P 1 . Yet this relationship between
matrix product and vector inner-product does not hold for general symmetric
projector matrices.
E.10 For diagonalizable X ∈ R m×m (A.5), its orthogonal projection in isomorphic R m2 on
the range of vectorized yz T ∈ R m×m becomes:
P 1 XP 2 =
m∑
λ i P 1 s i wi T P 2
i=1
When R(P 1 ) = R(w j ) and R(P 2 ) = R(s j ), the j th dyad term from the diagonalization
is isolated but only, in general, to within a scale factor because neither set of left or
right eigenvectors is necessarily orthonormal unless X is normal [344,3.2]. Yet when
R(P 2 )= R(s k ) , k≠j ∈{1... m}, then P 1 XP 2 =0.