v2009.01.01 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 117
yy T
a
I θ a
M
R
M 11T
Figure 41: Illustrated is a section, perpendicular to axis of revolution, of
circular cone from Figure 40. Radius R is distance from any extreme
direction to axis at a I . Vector a
M M 11T is an arbitrary reference by which
to measure angle θ .
orthogonal to the identity matrix I ;
〈C − a M I , I〉 = tr(C − a I) = 0 (217)
M
Proof. Although the positive semidefinite cone possesses some
characteristics of a circular cone, we can show it is not by demonstrating a
shortage of extreme directions; id est, some extreme directions corresponding
to each and every angle of rotation about the axis of revolution are
nonexistent: Referring to Figure 41, [332,1-7]
cos θ =
Solving for vector y we get
〈 a
M 11T − a M I , yyT − a M I〉
a 2 (1 − 1 M ) (218)
a(1 + (M −1) cosθ) = (1 T y) 2 (219)
Because this does not have real solution for every matrix dimension M and
for all 0 ≤ θ ≤ 2π , then we can conclude that the positive semidefinite cone
might be circular but only in matrix dimensions 1 and 2 . 2.39
2.39 In fact, the positive semidefinite cone is circular in matrix dimensions 1 and 2 while