v2009.01.01 - Convex Optimization

662 APPENDIX E. PROJECTION
E.6.2
Nonorthogonal projection on vectorized matrix
Formula (1837) is extensible. Given X,Y,Z ∈ R m×n , we have the
one-dimensional nonorthogonal projection of X in isomorphic R mn on the
range of vectorized Y : (2.2)
〈Z , X〉
Y , 〈Z , Y 〉 ≠ 0 (1839)
〈Z , Y 〉
where 〈Z , X〉/〈Z , Y 〉 is the coefficient of projection. The inequality
accounts for the fact: projection on R(vec Y ) is in a direction orthogonal to
vec Z .
E.6.2.1
Nonorthogonal projection on dyad
Now suppose we have nonorthogonal projector dyad
Analogous to (1837), for X ∈ R m×m
P 0 = yzT
z T y ∈ Rm×m (1840)
P 0 XP 0 =
yzT
z T y X yzT
z T y = zT Xy
(z T y) 2 yzT = 〈zyT , X〉
〈zy T , yz T 〉 yzT (1841)
is a nonorthogonal projection of matrix X on the range of vectorized
dyad P 0 ; from which it follows:
P 0 XP 0 = zT Xy
z T y
〈 〉
yz T zy
T yz
T
z T y = z T y , X z T y = 〈P 0 T , X〉 P 0 = 〈P 0 T , X〉
〈P0 T , P 0 〉 P 0
(1842)
Yet this relationship between matrix product and vector inner-product only
holds for a dyad projector. When nonsymmetric projector P 0 is rank-one as
in (1840), therefore,
R(vecP 0 XP 0 ) = R(vec P 0 ) in R m2 (1843)
and
P 0 XP 0 − X ⊥ P T 0 in R m2 (1844)