v2009.01.01 - Convex Optimization

E.6. VECTORIZATION INTERPRETATION, 667
which remains an orthonormal basis for S M . Then remarkably
∑
C 1 = M 〈QE ij Q T , C 1 〉 QE ij Q T
i,j=1
j ≥ i
∑
= M ∑
〈q i qi T , C 1 〉 q i qi T + M 〈QE ij Q T , QΛQ T 〉 QE ij Q T
i=1
∑
= M 〈q i qi T , C 1 〉 q i qi
T
i=1
i,j=1
j > i
∆ ∑
= M ∑
〈P i , C 1 〉 P i = M q i qi T C 1 q i qi
T
i=1
∑
= M λ i q i qi
T
i=1
i=1
∑
= M P i C 1 P i
i=1
(1859)
this orthogonal expansion becomes the diagonalization; still a sum of
one-dimensional orthogonal projections. The eigenvalues
λ i = 〈q i q T i , C 1 〉 (1860)
are clearly coefficients of projection of C 1 on the range of each vectorized
eigenmatrix. (conferE.6.2.1.1) The remaining M(M −1)/2 coefficients
(i≠j) are zeroed by projection. When P i is rank-one symmetric as in (1859),
R(svec P i C 1 P i ) = R(svec q i q T i ) = R(svec P i ) in R M(M+1)/2 (1861)
and
P i C 1 P i − C 1 ⊥ P i in R M(M+1)/2 (1862)
E.6.4.2
Positive semidefiniteness test as orthogonal projection
For any given X ∈ R m×m the familiar quadratic construct y T Xy ≥ 0,
over broad domain, is a fundamental test for positive semidefiniteness.
(A.2) It is a fact that y T Xy is always proportional to a coefficient of
orthogonal projection; letting z in formula (1851) become y ∈ R m , then
P 2 =P 1 =yy T /y T y=yy T /‖yy T ‖ 2 (confer (1496)) and formula (1852) becomes
〈yy T , X〉
〈yy T , yy T 〉 yyT = yT Xy
y T y
yy T
y T y = yyT
y T y X yyT
y T y
∆
= P 1 XP 1 (1863)