v2009.01.01 - Convex Optimization

546 APPENDIX A. LINEAR ALGEBRA
A.3.0.0.1 Theorem. Positive (semi)definite matrix.
A∈ S M is positive semidefinite if and only if for each and every vector x∈ R M
of unit norm, ‖x‖ = 1 , A.7 we have x T Ax ≥ 0 (1334);
A ≽ 0 ⇔ tr(xx T A) = x T Ax ≥ 0 ∀xx T (1343)
Matrix A ∈ S M is positive definite if and only if for each and every ‖x‖ = 1
we have x T Ax > 0 ;
A ≻ 0 ⇔ tr(xx T A) = x T Ax > 0 ∀xx T , xx T ≠ 0 (1344)
Proof. Statements (1343) and (1344) are each a particular instance
of dual generalized inequalities (2.13.2) with respect to the positive
semidefinite cone; videlicet, [313]
⋄
A ≽ 0 ⇔ 〈xx T , A〉 ≥ 0 ∀xx T (≽ 0)
A ≻ 0 ⇔ 〈xx T , A〉 > 0 ∀xx T (≽ 0), xx T ≠ 0
(1345)
This says: positive semidefinite matrix A must belong to the normal side
of every hyperplane whose normal is an extreme direction of the positive
semidefinite cone. Relations (1343) and (1344) remain true when xx T is
replaced with “for each and every” positive semidefinite matrix X ∈ S M +
(2.13.5) of unit norm, ‖X‖= 1, as in
A ≽ 0 ⇔ tr(XA) ≥ 0 ∀X ∈ S M +
A ≻ 0 ⇔ tr(XA) > 0 ∀X ∈ S M + , X ≠ 0
(1346)
But that condition is more than what is necessary. By the discretized
membership theorem in2.13.4.2.1, the extreme directions xx T of the positive
semidefinite cone constitute a minimal set of generators necessary and
sufficient for discretization of dual generalized inequalities (1346) certifying
membership to that cone.
A.7 The traditional condition requiring all x∈ R M for defining positive (semi)definiteness
is actually more than what is necessary. The set of norm-1 vectors is necessary and
sufficient to establish positive semidefiniteness; actually, any particular norm and any
nonzero norm-constant will work.