v2007.09.13 - Convex Optimization

A.7. ZEROS 515A.7.5.0.1 Proposition. (Sturm) Dyad-decompositions. [252,5.2]Let positive semidefinite matrix X ∈ S M + have rank ρ . Then given symmetricmatrix A∈ S M , 〈A, X〉 = 0 if and only if there exists a dyad-decompositionsatisfyingX =ρ∑x j xj T (1382)j=1〈A , x j x T j 〉 = 0 for each and every j ∈ {1... ρ} (1383)⋄The dyad-decomposition of X proposed is generally not that obtainedfrom a standard diagonalization by eigen decomposition, unless ρ =1 orthe given matrix A is diagonalizable simultaneously (A.7.4) with X .That means, elemental dyads in decomposition (1382) constitute a generallynonorthogonal set. Sturm & Zhang give a simple procedure for constructingthe dyad-decomposition; (F.5) matrix A may be regarded as a parameter.A.7.5.0.2 Example. Dyad.The dyad uv T ∈ R M×M (B.1) is zero definite on all x for which eitherx T u=0 or x T v=0;{x | x T uv T x = 0} = {x | x T u=0} ∪ {x | v T x=0} (1384)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} (1385)