v2009.01.01 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 105
The only symmetric positive semidefinite matrix in S M +
0-eigenvalues resides at the origin. (A.7.3.0.1)
having M
2.9.0.1 Membership
Observe the notation A ≽ 0 ; meaning (confer2.3.1.1), matrix A belongs to
the positive semidefinite cone in the subspace of symmetric matrices, whereas
A ≻ 0 denotes membership to that cone’s interior. (2.13.2) Notation A ≻ 0
can be read: symmetric matrix A is greater than zero with respect to the
positive semidefinite cone. This notation further implies that coordinates
[sic] for orthogonal expansion of a positive (semi)definite matrix must be its
(nonnegative) positive eigenvalues (2.13.7.1.1,E.6.4.1.1) when expanded in
its eigenmatrices (A.5.1).
Generalizing comparison on the real line, the notation A ≽B denotes
comparison with respect to the positive semidefinite cone; (A.3.1) id est,
A ≽B ⇔ A −B ∈ S M + but neither matrix A or B necessarily belongs to
the positive semidefinite cone. Yet, (1387) A ≽B , B ≽0 ⇒ A≽0 ; id est,
A ∈ S M + . (confer Figure 55)
2.9.0.1.1 Example. Equality constraints in semidefinite program (584).
Employing properties of partial ordering (2.7.2.2) for the pointed closed
convex positive semidefinite cone, it is easy to show, given A + S = C
S ≽ 0 ⇔ A ≼ C (176)
2.9.1 Positive semidefinite cone is convex
The set of all positive semidefinite matrices forms a convex cone in the
ambient space of symmetric matrices because any pair satisfies definition
(157); [176,7.1] videlicet, for all ζ 1 , ζ 2 ≥ 0 and each and every A 1 , A 2 ∈ S M
ζ 1 A 1 + ζ 2 A 2 ≽ 0 ⇐ A 1 ≽ 0, A 2 ≽ 0 (177)
a fact easily verified by the definitive test for positive semidefiniteness of a
symmetric matrix (A):
A ≽ 0 ⇔ x T Ax ≥ 0 for each and every ‖x‖ = 1 (178)