v2010.10.26 - Convex Optimization

600 APPENDIX A. LINEAR ALGEBRAA.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 Mof unit norm, ‖x‖ = 1 , A.7 we have x T Ax ≥ 0 (1431);A ≽ 0 ⇔ tr(xx T A) = x T Ax ≥ 0 ∀xx T (1440)Matrix A ∈ S M is positive definite if and only if for each and every ‖x‖ = 1we have x T Ax > 0 ;A ≻ 0 ⇔ tr(xx T A) = x T Ax > 0 ∀xx T , xx T ≠ 0 (1441)Proof. Statements (1440) and (1441) are each a particular instanceof dual generalized inequalities (2.13.2) with respect to the positivesemidefinite cone; videlicet, [360]⋄A ≽ 0 ⇔ 〈xx T , A〉 ≥ 0 ∀xx T (≽ 0)A ≻ 0 ⇔ 〈xx T , A〉 > 0 ∀xx T (≽ 0), xx T ≠ 0(1442)This says: positive semidefinite matrix A must belong to the normal sideof every hyperplane whose normal is an extreme direction of the positivesemidefinite cone. Relations (1440) and (1441) remain true when xx T isreplaced with “for each and every” positive semidefinite matrix X ∈ S M +(2.13.5) of unit norm, ‖X‖= 1, as inA ≽ 0 ⇔ tr(XA) ≥ 0 ∀X ∈ S M +A ≻ 0 ⇔ tr(XA) > 0 ∀X ∈ S M + , X ≠ 0(1443)But that condition is more than what is necessary. By the discretizedmembership theorem in2.13.4.2.1, the extreme directions xx T of the positivesemidefinite cone constitute a minimal set of generators necessary andsufficient for discretization of dual generalized inequalities (1443) certifyingmembership to that cone.A.7 The traditional condition requiring all x∈ R M for defining positive (semi)definitenessis actually more than what is necessary. The set of norm-1 vectors is necessary andsufficient to establish positive semidefiniteness; actually, any particular norm and anynonzero norm-constant will work.