v2010.10.26 - Convex Optimization

610 APPENDIX A. LINEAR ALGEBRAWe can deduce from these, given nonsingular matrix Z and any particulardimensionally[ ]compatible Y : matrix A∈ S M is positive semidefinite if andZTonly ifY T A[Z Y ] is positive semidefinite. In other words, from theCorollary it follows: for dimensionally compatible ZA ≽ 0 ⇔ Z T AZ ≽ 0 and Z T has a left inverseProducts such as Z † Z and ZZ † are symmetric and positive semidefinitealthough, given A ≽ 0, Z † AZ and ZAZ † are neither necessarily symmetricor positive semidefinite.A.3.1.0.6 Theorem. Symmetric projector semidefinite. [20,III][21,6] [222, p.55] For symmetric idempotent matrices P and RP,R ≽ 0P ≽ R ⇔ R(P ) ⊇ R(R) ⇔ N(P ) ⊆ N(R)(1509)Projector P is never positive definite [333,6.5 prob.20] unless it is theidentity matrix.⋄A.3.1.0.7 Theorem. Symmetric positive semidefinite. [202, p.400]Given real matrix Ψ with rank Ψ = 1Ψ ≽ 0 ⇔ Ψ = uu T (1510)where u is some real vector; id est, symmetry is necessary and sufficient forpositive semidefiniteness of a rank-1 matrix.⋄Proof. Any rank-one matrix must have the form Ψ = uv T . (B.1)Suppose Ψ is symmetric; id est, v = u . For all y ∈ R M , y T uu T y ≥ 0.Conversely, suppose uv T is positive semidefinite. We know that can hold ifand only if uv T + vu T ≽ 0 ⇔ for all normalized y ∈ R M , 2y T uv T y ≥ 0 ;but that is possible only if v = u .The same does not hold true for matrices of higher rank, as Example A.2.1.0.1shows.