v2006.03.09 - Convex Optimization

460 APPENDIX A. LINEAR ALGEBRAThe positive semidefinite matrix[ ] 1 2A =0 1for example, has no nullspace. Yet(1185){x | x T Ax = 0} = {x | 1 T x = 0} ⊂ R 2 (1186)which is the nullspace of the symmetrized matrix. Symmetric matrices arenot spared from the excess; videlicet,[ ] 1 2B =(1187)2 1has eigenvalues {−1, 3}, no nullspace, but is zero definite on A.19X ∆ = {x∈ R 2 | x 2 = (−2 ± √ 3)x 1 } (1188)A.7.4.0.1 Proposition. (Sturm) Dyad-decompositions. [217,5.2]Given symmetric matrix A ∈ S M , let positive semidefinite matrix X ∈ S M +have rank ρ . Then 〈A, X〉=0 if and only if there is a dyad-decompositionρ∑X = x j xj T (1189)satisfyingj=1〈A, x j x T j 〉 = 〈A, X〉 = 0 for each and every j (1190)⋄A.7.4.0.2 Example. Dyad.The dyad uv T ∈ R M×M (B.1) is zero definite on all x for which either x T u=0or x T v=0;{x | x T uv T x = 0} = {x | x T u=0} ∪ {x | v T x=0} (1191)id est, on u ⊥ ∪ v ⊥ . Symmetrizing the dyad does not change the outcome:{x | x T (uv T + vu T )x/2 = 0} = {x | x T u=0} ∪ {x | v T x=0} (1192)A.19 These two lines represent the limit in the union of two generally distinct hyperbolae;id est, for matrix B and set X as definedlim{x∈ R 2 | x T Bx = ε} = Xε↓0