v2009.01.01 - Convex Optimization

668 APPENDIX E. PROJECTION
By (1850), product P 1 XP 1 is the one-dimensional orthogonal projection of
X in isomorphic R m2 on the range of vectorized P 1 because, for rankP 1 =1
and P 2
1 =P 1 ∈ S m (confer (1842))
P 1 XP 1 = yT Xy
y T y
〈 〉
yy T yy
T yy
T
y T y = y T y , X y T y = 〈P 1 , X〉 P 1 = 〈P 1 , X〉
〈P 1 , P 1 〉 P 1
(1864)
The coefficient of orthogonal projection 〈P 1 , X〉= y T Xy/(y T y) is also known
as Rayleigh’s quotient. E.12 When P 1 is rank-one symmetric as in (1863),
R(vecP 1 XP 1 ) = R(vec P 1 ) in R m2 (1865)
and
P 1 XP 1 − X ⊥ P 1 in R m2 (1866)
The test for positive semidefiniteness, then, is a test for nonnegativity of
the coefficient of orthogonal projection of X on the range of each and every
vectorized extreme direction yy T (2.8.1) from the positive semidefinite cone
in the ambient space of symmetric matrices.
E.12 When y becomes the j th eigenvector s j of diagonalizable X , for example, 〈P 1 , X 〉
becomes the j th eigenvalue: [170,III]
〈P 1 , X 〉| y=sj
=
s T j
( m∑
λ i s i wi
T
i=1
s T j s j
)
s j
= λ j
Similarly for y = w j , the j th left-eigenvector,
〈P 1 , X 〉| y=wj
=
w T j
( m∑
λ i s i wi
T
i=1
w T j w j
)
w j
= λ j
A quandary may arise regarding the potential annihilation of the antisymmetric part of
X when s T j Xs j is formed. Were annihilation to occur, it would imply the eigenvalue thus
found came instead from the symmetric part of X . The quandary is resolved recognizing
that diagonalization of real X admits complex eigenvectors; hence, annihilation could only
come about by forming Re(s H j Xs j) = s H j (X +XT )s j /2 [176,7.1] where (X +X T )/2 is
the symmetric part of X , and s H j denotes conjugate transpose.